#6936922

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884841
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	#38903	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1587 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=f,d^2=fh,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,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, 6844118);