#6637209

order	512
nilpotency class	3
isoclinism	isoclinic class of #6636908
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 ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1446 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)(9,10)(11,13)(12,14)(15,16)(17,18,23,27)(19,24,31,40)(20,32,34,41)(21,26,22,36)(25,35,42,33)(28,43,46,47)(29,38,30,39)(37,45,48,44);(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)(32,43)(33,44)(34,46)(35,45)(36,39)(41,47)(42,48);(11,15)(13,16)(17,20,21,35,23,34,22,33)(18,25,36,41,27,42,26,32)(19,28,29,45,31,46,30,44)(24,37,39,47,40,48,38,43);(2,6)(3,8)(11,15)(13,16)(17,21,23,22)(18,26,27,36)(19,29,31,30)(20,33,34,35)(24,38,40,39)(25,41,42,32)(28,44,46,45)(37,47,48,43);(2,6)(5,7)(10,14)(13,16)(17,21,23,22)(18,26,27,36)(19,30,31,29)(20,33,34,35)(24,39,40,38)(25,41,42,32)(28,45,46,44)(37,43,48,47)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6544405);