#6799584

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1280 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,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, 6706780);