#7017539

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1266 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=fi,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh,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, 6924735);