#421043

order	512
nilpotency class	3
isoclinism	isoclinic class of #421042
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15283
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

1812 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318867, #72318925, #72318981
maximal normal subgroup of	#72318867, #72318925, #72318981

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 328239);