#7010134

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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

1354 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)(11,13)(12,14)(15,16)(17,18,21,26,22,27,37,44)(19,28,31,46,32,23,38,41)(20,33,36,42,35,25,34,24)(29,43,48,39,30,40,47,45);(1,3,5,8)(2,6,4,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,23)(20,29)(21,31)(22,32)(24,39)(25,43)(26,41)(27,28)(30,35)(33,40)(34,47)(36,48)(37,38)(42,45)(44,46);(1,2,5,4)(3,6,8,7)(9,11)(10,13)(12,16)(14,15)(17,20)(18,24)(19,29)(21,34)(22,35)(23,39)(25,26)(27,42)(28,45)(30,32)(31,47)(33,44)(36,37)(38,48)(40,46)(41,43);(1,4,5,2)(3,6,8,7)(11,16)(13,15)(17,20)(18,25)(19,30)(21,36)(22,35)(23,40)(24,26)(27,33)(28,43)(29,32)(31,47)(34,37)(38,48)(39,46)(41,45)(42,44);(10,14)(13,15)(17,21,22,37)(18,26,27,44)(19,31,32,38)(20,34,35,36)(23,41,28,46)(24,25,42,33)(29,47,30,48)(39,43,45,40)

Polycyclic presentation:

2^9|a^2=fgi,b^2=fi,c^2=fi,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,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

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6917330);