#6772597

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1489 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6679793);