#7055339

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	#42910	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1755 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=h,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,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, 6962535);