#6797591

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
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	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1781 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)(9,10)(11,15)(12,14)(13,16)(17,18,23,27)(19,24,22,37)(20,30,33,47)(21,26,36,43)(25,28,40,32)(29,38,35,42)(31,46,34,48)(39,44,41,45);(2,5)(3,7)(9,11)(10,13)(12,16)(14,15)(17,19,23,22)(18,24,27,37)(20,28,33,32)(21,29,36,35)(25,30,40,47)(26,38,43,42)(31,44,34,45)(39,46,41,48);(11,16)(13,15)(17,20,19,28,23,33,22,32)(18,25,37,47,27,40,24,30)(21,34,29,45,36,31,35,44)(26,41,42,46,43,39,38,48);(2,5)(6,8)(10,14)(13,15)(17,21,23,36)(18,26,27,43)(19,29,22,35)(20,31,33,34)(24,38,37,42)(25,39,40,41)(28,44,32,45)(30,46,47,48);(1,3,4,7)(2,6,5,8)(17,19,23,22)(18,24,27,37)(20,28,33,32)(21,35,36,29)(25,30,40,47)(26,42,43,38)(31,45,34,44)(39,48,41,46)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6704787);