#6772980

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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	C₂×#121	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1579 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6680176);