#6473439

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470860
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1502 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=gi,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,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, 6380635);