#6433650

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432861
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 ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1690 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,28)(19,24,31,40)(20,32,35,44)(21,26,36,41)(22,37,30,27)(25,38,43,34)(29,42,45,48)(33,47,46,39);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,23,31)(18,24,28,40)(20,29,35,45)(21,30,36,22)(25,39,43,47)(26,27,41,37)(32,42,44,48)(33,34,46,38);(11,15)(13,16)(17,20,23,35)(18,25,28,43)(19,29,31,45)(21,33,36,46)(22,38,30,34)(24,39,40,47)(26,42,41,48)(27,44,37,32);(10,14)(13,16)(17,21)(18,26)(19,30)(20,33)(22,31)(23,36)(24,27)(25,42)(28,41)(29,34)(32,47)(35,46)(37,40)(38,45)(39,44)(43,48);(1,3,4,7)(2,6,5,8)(19,31)(21,36)(24,40)(26,41)(29,45)(33,46)(39,47)(42,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,c^b=ch,e^b=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6340846);