#6854212

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1399 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)(3,7)(4,6)(5,8)(17,18,22,28)(19,27,23,24)(20,31,34,39)(21,35,38,26)(25,29,42,32)(30,40,37,44)(33,48,36,47)(41,46,43,45);(1,3)(2,5)(4,8)(6,7)(9,10,13,16)(11,15,12,14)(17,19,22,23)(18,24,28,27)(20,32,34,29)(21,30,38,37)(25,31,42,39)(26,40,35,44)(33,46,36,45)(41,47,43,48);(2,6)(3,8)(9,11,13,12)(10,14,16,15)(17,20,23,32,22,34,19,29)(18,25,27,31,28,42,24,39)(21,36,37,45,38,33,30,46)(26,43,44,48,35,41,40,47);(2,6)(5,7)(17,21,22,38)(18,26,28,35)(19,30,23,37)(20,33,34,36)(24,40,27,44)(25,41,42,43)(29,45,32,46)(31,47,39,48);(9,12,13,11)(10,15,16,14)(17,19,22,23)(18,27,28,24)(20,29,34,32)(21,37,38,30)(25,31,42,39)(26,40,35,44)(33,46,36,45)(41,48,43,47)

Polycyclic presentation:

2^9|a^2=i,b^2=fhi,c^2=ghi,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, 6761408);