#6912729

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
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×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1634 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=g,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,d^c=dfh,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6819925);