#6750060

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682460
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

1662 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)(19,24)(20,32)(21,26)(22,36)(23,27)(25,33)(28,43)(29,38)(30,39)(31,40)(34,42)(35,41)(37,45)(44,47)(46,48);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,28)(21,30)(22,29)(23,31)(25,37)(26,39)(27,40)(32,43)(33,45)(34,44)(35,46)(36,38)(41,48)(42,47);(11,15)(13,16)(17,20,21,34,23,35,22,33)(18,25,36,41,27,42,26,32)(19,28,30,44,31,46,29,45)(24,37,38,48,40,47,39,43);(2,6)(5,7)(17,21,23,22)(18,26,27,36)(19,29,31,30)(20,33,35,34)(24,38,40,39)(25,41,42,32)(28,44,46,45)(37,43,47,48);(3,8)(5,7)(11,15)(13,16)(17,21,23,22)(18,26,27,36)(19,30,31,29)(20,34,35,33)(24,39,40,38)(25,32,42,41)(28,44,46,45)(37,43,47,48)

Polycyclic presentation:

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

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, 6657256);