#6793797

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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1496 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)(17,18,23,24,22,27,19,28)(20,31,29,41,34,38,32,25)(21,35,37,42,36,26,30,39)(33,46,43,40,47,44,45,48);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,13)(14,16)(17,19,22,23)(18,24,27,28)(20,32,34,29)(21,30,36,37)(25,31,41,38)(26,39,35,42)(33,45,47,43)(40,44,48,46);(10,13)(14,15)(17,20,22,34)(18,25,27,41)(19,29,23,32)(21,33,36,47)(24,38,28,31)(26,40,35,48)(30,43,37,45)(39,46,42,44);(2,6)(5,7)(17,21)(18,26)(19,30)(20,33)(22,36)(23,37)(24,39)(25,40)(27,35)(28,42)(29,43)(31,44)(32,45)(34,47)(38,46)(41,48);(3,8)(5,7)(9,11,12,16)(10,14,13,15)(21,36)(26,35)(30,37)(33,47)(39,42)(40,48)(43,45)(44,46)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,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, 6700993);