#451784

order	512
nilpotency class	3
isoclinism	isoclinic class of #451783
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16014
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

932 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 932 conjugacy classes of subgroups — too many to draw. Only its 168 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,6,13,7,14,8,12)(3,15,30,35,19,34,18,9)(4,20,23,39,24,38,25,10)(5,26,28,42,29,41,31,11)(16,36,49,43,21,32,46,58)(17,33,47,40,48,56,27,44)(22,37,53,61,54,62,55,50)(45,60,52,57,51,63,64,59);(1,3)(2,9)(4,21)(5,27)(6,30)(7,19)(8,18)(10,36)(11,40)(12,34)(13,15)(14,35)(16,24)(17,28)(20,43)(22,52)(23,46)(25,49)(26,56)(29,47)(31,48)(32,39)(33,41)(37,60)(38,58)(42,44)(45,55)(50,59)(51,53)(54,64)(57,61)(62,63);(1,4,7,24)(2,10,14,39)(3,16,19,21)(5,22,29,54)(6,25,8,23)(9,32,35,36)(11,37,42,62)(12,20,13,38)(15,43,34,58)(17,45,48,51)(18,49,30,46)(26,50,41,61)(27,52,47,64)(28,55,31,53)(33,57,56,59)(40,60,44,63);(1,5,7,29)(2,11)(3,17,19,48)(4,22,24,54)(6,31,8,28)(9,33)(10,37)(12,26)(13,41)(14,42)(15,44)(16,45,21,51)(18,47,30,27)(20,50)(23,55,25,53)(32,57)(34,40)(35,56)(36,59)(38,61)(39,62)(43,63)(46,52,49,64)(58,60)

Polycyclic presentation:

2^9|a^2=gh,c^2=i,d^2=ef,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317787, #72318497, #72318589
maximal normal subgroup of	#72317787, #72318497, #72318589

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 358980);