#7127035

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1286 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)(9,10,14,11)(12,15,13,16)(17,18)(19,23)(20,25)(21,24)(22,27)(26,33)(28,35)(29,34)(30,37)(31,40)(32,38)(36,43)(39,44)(41,47)(42,45)(46,48);(1,3)(2,6)(4,7)(5,8)(10,11)(12,13)(17,19,22,30)(18,23,27,37)(20,28,31,41)(21,29,33,43)(24,34,26,36)(25,35,40,47)(32,42,44,48)(38,45,39,46);(9,11,14,10)(12,15,13,16)(18,24)(20,31)(21,33)(23,34)(25,39)(26,27)(28,41)(29,43)(35,46)(36,37)(38,40)(45,47);(3,8)(6,7)(9,12,14,13)(10,15,11,16)(17,20)(18,25)(19,28)(21,32)(22,31)(23,35)(24,38)(26,39)(27,40)(29,42)(30,41)(33,44)(34,45)(36,46)(37,47)(43,48);(1,4,5,2)(3,7,8,6)(9,13,14,12)(10,15,11,16)(20,31)(25,40)(28,41)(32,44)(35,47)(38,39)(42,48)(45,46)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,d^2=h,e^2=f,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,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, 7034231);