#6766710

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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1582 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)(3,7)(4,6)(5,8)(17,18,22,27)(19,23,31,40)(20,25,32,43)(21,34,36,26)(24,35,42,33)(28,44,29,38)(30,47,48,39)(37,46,41,45);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,29)(21,30)(22,31)(24,41)(25,38)(26,39)(27,40)(28,32)(33,46)(34,47)(35,45)(36,48)(37,42)(43,44);(10,13)(14,16)(17,20)(18,24)(19,28)(21,35)(22,32)(23,37)(25,34)(26,43)(27,42)(29,31)(30,46)(33,36)(38,39)(40,41)(44,47)(45,48);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,20)(18,25)(19,29)(21,33)(22,32)(23,38)(24,34)(26,42)(27,43)(28,31)(30,46)(35,36)(37,39)(40,44)(41,47)(45,48);(3,8)(5,7)(10,13)(14,16)(20,32)(24,42)(25,43)(28,29)(33,35)(37,41)(38,44)(45,46)

Polycyclic presentation:

2^9|a^2=i,d^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,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, 6673906);