#6992817

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	#39131	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1506 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fhi,d^2=fhi,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,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, 6900013);