#6435996

order	512
nilpotency class	3
isoclinism	isoclinic class of #6433044
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1434 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:

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

Polycyclic presentation:

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

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, 6343192);