#6749957

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
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×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1658 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=g,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6657153);