#6963623

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884841
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	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1780 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)(3,4)(5,6)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18,23,27)(19,24,22,35)(20,30,32,37)(21,26,33,41)(25,36,40,28)(29,38,34,45)(31,43,44,47)(39,46,48,42);(2,6)(4,7)(9,11)(10,13)(12,16)(14,15)(17,19,23,22)(18,24,27,35)(20,28,32,36)(21,29,33,34)(25,37,40,30)(26,38,41,45)(31,42,44,46)(39,47,48,43);(1,3)(2,7)(4,6)(5,8)(11,16)(13,15)(17,20,23,32)(18,25,27,40)(19,28,22,36)(21,31,33,44)(24,37,35,30)(26,39,41,48)(29,42,34,46)(38,47,45,43);(2,6)(3,8)(10,14)(13,15)(17,21,23,33)(18,26,27,41)(19,29,22,34)(20,31,32,44)(24,38,35,45)(25,39,40,48)(28,42,36,46)(30,43,37,47);(1,4,5,7)(2,3,6,8)(18,27)(21,33)(24,35)(25,40)(29,34)(30,37)(31,44)(42,46)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6870819);