#6633324

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632088
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#25733	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1651 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)(3,5)(4,6)(7,8)(10,14)(11,12)(17,18)(19,25)(20,32)(21,27)(22,28)(23,37)(24,29)(26,34)(30,45)(31,40)(33,48)(35,43)(36,41)(38,44)(39,47)(42,46);(1,3)(2,5)(4,7)(6,8)(9,10,13,14)(11,16,12,15)(17,19)(18,25)(20,30)(21,24)(22,36)(23,31)(26,39)(27,29)(28,41)(32,45)(33,35)(34,47)(37,40)(38,46)(42,44)(43,48);(3,7)(5,8)(9,11,13,12)(10,15,14,16)(17,20)(18,26)(19,30)(21,33)(22,34)(23,38)(24,35)(25,39)(27,42)(28,43)(29,44)(31,46)(32,37)(36,47)(40,45)(41,48);(3,7)(5,8)(9,12,13,11)(10,15,14,16)(17,21,24,19)(18,27,29,25)(20,33,35,30)(22,31,23,36)(26,42,44,39)(28,40,37,41)(32,48,43,45)(34,46,38,47);(2,6)(5,8)(17,22,24,23)(18,28,29,37)(19,31,21,36)(20,34,35,38)(25,40,27,41)(26,43,44,32)(30,46,33,47)(39,45,42,48)

Polycyclic presentation:

2^9|b^2=f,c^2=f,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfhi,e^b=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6540520);