#7008005

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1323 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)(17,18,23,36,21,24,22,27)(19,28,35,25,29,43,31,38)(20,26,37,48,33,39,34,42)(30,44,47,40,32,46,45,41);(1,3,5,8)(2,6,4,7)(9,10)(11,15)(12,14)(13,16)(18,24)(19,29)(26,39)(27,36)(30,32)(31,35)(42,48)(45,47);(9,11)(10,13)(12,16)(14,15)(17,19,21,29)(18,25,24,38)(20,32,33,30)(22,35,23,31)(26,41,39,40)(27,43,36,28)(34,45,37,47)(42,44,48,46);(1,4,5,2)(3,6,8,7)(11,16)(13,15)(17,20)(18,26)(19,30)(21,33)(22,34)(23,37)(24,39)(25,40)(27,42)(28,44)(29,32)(31,45)(35,47)(36,48)(38,41)(43,46);(10,14)(13,15)(18,24)(20,33)(25,38)(27,36)(28,43)(30,32)(34,37)(45,47)

Polycyclic presentation:

2^9|a^2=fg,b^2=fi,c^2=i,d^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dfi,e^b=ehi,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, 6915201);