#7043513

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1250 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=fi,c^2=gi,d^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6950709);