#6536634

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535815
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#24397	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1803 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,6)(4,5)(7,8)(17,18)(19,28)(20,32)(21,26)(22,37)(23,27)(24,31)(25,34)(29,44)(30,40)(33,42)(35,39)(36,41)(38,46)(43,45)(47,48);(9,10)(11,14)(12,13)(15,16)(17,19,22,31,23,30,21,35)(18,24,26,28,27,39,37,40)(20,29,33,47,34,45,36,46)(25,38,42,43,32,48,41,44);(2,5)(6,8)(10,13)(14,16)(17,20)(18,25)(19,29)(21,36)(22,33)(23,34)(24,38)(26,42)(27,32)(28,43)(30,45)(31,47)(35,46)(37,41)(39,48)(40,44);(1,3,4,7)(2,6,5,8)(9,11)(10,14)(12,15)(13,16)(19,30)(24,39)(28,40)(29,45)(31,35)(38,48)(43,44)(46,47);(11,15)(14,16)(17,21,23,22)(18,26,27,37)(19,31,30,35)(20,33,34,36)(24,40,39,28)(25,41,32,42)(29,46,45,47)(38,43,48,44)

Polycyclic presentation:

2^9|b^2=f,d^2=gi,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=ei,e^c=ei,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6443830);