#395067

order	512
nilpotency class	3
isoclinism	isoclinic class of #394113
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1450 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 72 points:

(1,2)(3,6)(4,5)(7,8)(9,10,16,22)(11,17,26,41)(12,27,32,44)(13,33,36,46)(14,21,15,20)(18,31,45,30)(19,34,47,35)(23,42,50,67)(24,51,54,65)(25,48,37,40)(28,38,55,64)(29,58,61,43)(39,53,66,52)(49,63,70,72)(56,59,62,60)(57,71,69,68);(2,5)(6,8)(9,11)(10,17)(12,28)(13,24)(14,37)(15,25)(16,26)(18,42)(19,39)(20,48)(21,40)(22,41)(23,30)(27,38)(29,57)(31,50)(32,55)(33,51)(34,53)(35,52)(36,54)(43,68)(44,64)(45,67)(46,65)(47,66)(49,60)(56,63)(58,71)(59,70)(61,69)(62,72);(9,12,16,32)(10,18,22,45)(11,23,26,50)(13,29,36,61)(14,31,15,30)(17,38,41,64)(19,43,47,58)(20,44,21,27)(24,49,54,70)(25,55,37,28)(33,62,46,56)(34,60,35,59)(39,63,66,72)(40,67,48,42)(51,68,65,71)(52,57,53,69);(1,3,4,7)(2,6,5,8)(9,13)(10,19)(11,24)(12,29)(14,35)(15,34)(16,36)(17,39)(18,43)(20,33)(21,46)(22,47)(23,49)(25,53)(26,54)(27,56)(28,57)(30,60)(31,59)(32,61)(37,52)(38,63)(40,65)(41,66)(42,68)(44,62)(45,58)(48,51)(50,70)(55,69)(64,72)(67,71)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 302263);