#6435646

order	512
nilpotency class	3
isoclinism	isoclinic class of #29602
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1528 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)(11,15)(14,16)(17,18,22,27,23,28,21,26)(19,24,32,40,30,38,31,39)(20,33,34,41,35,42,36,25)(29,43,45,47,44,48,46,37);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,29)(21,31)(22,32)(23,30)(25,37)(26,39)(27,40)(28,38)(33,43)(34,45)(35,44)(36,46)(41,47)(42,48);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,20)(18,25)(19,29)(21,34)(22,36)(23,35)(24,37)(26,33)(27,42)(28,41)(30,44)(31,45)(32,46)(38,47)(39,43)(40,48);(9,11,12,15)(10,14,13,16)(19,30)(24,38)(29,44)(31,32)(37,47)(39,40)(43,48)(45,46);(17,21,23,22)(18,26,28,27)(19,31,30,32)(20,34,35,36)(24,39,38,40)(25,33,41,42)(29,45,44,46)(37,43,47,48)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6342842);