#6908087

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
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	#37656	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2002 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)(11,15)(12,14)(13,16)(17,18,23,27)(19,24,22,36)(20,30,32,43)(21,26,34,40)(25,28,39,33)(29,37,35,46)(31,42,45,48)(38,41,47,44);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,16)(14,15)(17,19,23,22)(18,24,27,36)(20,28,32,33)(21,29,34,35)(25,30,39,43)(26,37,40,46)(31,41,45,44)(38,42,47,48);(1,2)(3,5)(4,6)(7,8)(11,16)(13,15)(17,20,19,28,23,32,22,33)(18,25,36,43,27,39,24,30)(21,31,29,41,34,45,35,44)(26,38,46,48,40,47,37,42);(2,6)(5,7)(10,14)(13,15)(17,21,23,34)(18,26,27,40)(19,29,22,35)(20,31,32,45)(24,37,36,46)(25,38,39,47)(28,41,33,44)(30,42,43,48);(3,8)(5,7)(20,32)(21,34)(25,39)(26,40)(28,33)(29,35)(30,43)(37,46)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gi,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfh,e^c=ei,e^d=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, 6815283);