#6997035

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	#2431	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1667 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)(6,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,27)(20,25)(21,23)(22,35)(24,30)(26,34)(28,43)(29,42)(31,38)(32,46)(33,48)(36,37)(39,44)(40,47)(41,45);(1,2,4,5)(3,7,8,6)(9,10)(11,15)(12,14)(13,16)(18,23)(20,31)(24,37)(26,35)(27,42)(28,32)(33,45)(44,47);(9,11)(10,13)(12,16)(14,15)(17,19,21,29)(18,24,23,37)(20,32,31,28)(22,36,34,30)(25,40,38,39)(26,42,35,27)(33,44,45,47)(41,43,48,46);(1,3,4,8)(2,6,5,7)(11,16)(13,15)(17,20,21,31)(18,25,23,38)(19,28,29,32)(22,33,34,45)(24,39,37,40)(26,41,35,48)(27,43,42,46)(30,44,36,47);(10,14)(13,15)(18,23)(20,31)(24,37)(26,35)(27,42)(28,32)(33,45)(44,47)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6904231);