#454662

order	512
nilpotency class	3
isoclinism	isoclinic class of #454661
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16255
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1751 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319622, #72319954, #72320118
maximal normal subgroup of	#72319622, #72319954, #72320118

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 361858);