#6969826

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884841
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	#2535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1376 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)(3,6)(4,8)(5,7)(10,14)(11,12)(17,18,23,25,22,28,21,27)(19,24,34,39,29,38,31,36)(20,26,35,40,32,42,33,41)(30,37,45,47,43,48,44,46);(1,2,6,3)(4,7,5,8)(9,10,13,14)(11,16,12,15)(17,19,22,29)(18,24,28,38)(20,30,32,43)(21,31,23,34)(25,39,27,36)(26,37,42,48)(33,44,35,45)(40,47,41,46);(1,3,6,2)(4,7,5,8)(9,11,13,12)(10,15,14,16)(18,25)(19,29)(20,32)(21,23)(24,36)(26,41)(27,28)(37,47)(38,39)(40,42)(44,45)(46,48);(1,4,6,5)(2,7,3,8)(9,12,13,11)(10,15,14,16)(17,20)(18,26)(19,30)(21,33)(22,32)(23,35)(24,37)(25,40)(27,41)(28,42)(29,43)(31,44)(34,45)(36,46)(38,48)(39,47);(1,5,6,4)(2,7,3,8)(19,29)(20,32)(24,38)(26,42)(31,34)(33,35)(36,39)(40,41)

Polycyclic presentation:

2^9|a^2=fgi,b^2=fi,c^2=f,d^2=f,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh,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, 6877022);