#6852556

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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 ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1449 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)(3,7)(4,6)(5,8)(17,18,22,28)(19,27,23,24)(20,31,34,39)(21,35,37,26)(25,32,41,29)(30,38,36,42)(33,46,47,44)(40,43,48,45);(1,3)(2,5)(4,8)(6,7)(9,10,13,16)(11,15,12,14)(17,19,22,23)(18,24,28,27)(20,32,34,29)(21,30,37,36)(25,39,41,31)(26,38,35,42)(33,45,47,43)(40,46,48,44);(3,8)(5,7)(9,11,13,12)(10,14,16,15)(17,20)(18,25)(19,29)(21,33)(22,34)(23,32)(24,31)(26,40)(27,39)(28,41)(30,43)(35,48)(36,45)(37,47)(38,44)(42,46);(2,6)(5,7)(17,21,22,37)(18,26,28,35)(19,30,23,36)(20,33,34,47)(24,38,27,42)(25,40,41,48)(29,43,32,45)(31,44,39,46);(9,12,13,11)(10,15,16,14)(17,19,22,23)(18,27,28,24)(20,29,34,32)(21,36,37,30)(25,39,41,31)(26,38,35,42)(33,45,47,43)(40,44,48,46)

Polycyclic presentation:

2^9|a^2=i,b^2=fhi,c^2=fh,d^2=i,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,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, 6759752);