#7061605

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

1404 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)(17,18,22,26)(19,23,28,36)(20,25,31,40)(21,24,33,27)(29,35,41,47)(30,37,42,34)(32,38,44,39)(43,46,48,45);(1,3,5,8)(2,6,4,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,23)(20,29)(21,30)(22,28)(24,37)(25,35)(26,36)(27,34)(31,41)(32,43)(33,42)(38,46)(39,45)(40,47)(44,48);(9,11,12,16)(10,13,14,15)(18,24)(19,28)(20,31)(21,33)(23,34)(25,39)(26,27)(35,46)(36,37)(38,40)(43,48)(45,47);(1,2,5,4)(3,7,8,6)(11,16)(13,15)(17,20,22,31)(18,25,26,40)(19,29,28,41)(21,32,33,44)(23,35,36,47)(24,38,27,39)(30,43,42,48)(34,45,37,46);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(18,26)(19,28)(20,31)(24,27)(30,42)(32,44)(35,47)(45,46)

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=hi,d^2=fi,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=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, 6968801);