#6966134

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884841
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

1560 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)(17,18,22,27,23,28,19,24)(20,31,29,40,33,39,36,25)(21,26,34,41,35,42,30,37)(32,44,43,47,46,48,45,38);(1,3)(2,5)(4,8)(6,7)(10,13)(11,15)(17,19,23,22)(18,24,28,27)(20,29,33,36)(21,30,35,34)(25,31,40,39)(26,37,42,41)(32,43,46,45)(38,44,47,48);(3,8)(5,7)(9,10)(11,14)(12,13)(15,16)(17,20,23,33)(18,25,28,40)(19,29,22,36)(21,32,35,46)(24,31,27,39)(26,38,42,47)(30,43,34,45)(37,44,41,48);(2,6)(5,7)(10,13)(14,16)(17,21)(18,26)(19,30)(20,32)(22,34)(23,35)(24,37)(25,38)(27,41)(28,42)(29,43)(31,44)(33,46)(36,45)(39,48)(40,47);(9,11,12,15)(10,14,13,16)(17,19,23,22)(18,27,28,24)(20,29,33,36)(21,34,35,30)(25,39,40,31)(26,37,42,41)(32,45,46,43)(38,44,47,48)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6873330);