#6809673

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806827
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#525	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1552 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)(3,9)(4,8)(5,7)(6,11)(10,16)(12,14)(13,15)(17,18)(19,24)(20,32)(21,34)(22,27)(23,28)(25,36)(26,37)(29,44)(30,43)(31,39)(33,48)(35,41)(38,47)(40,46)(42,45);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,19,23,31)(18,24,28,39)(20,22,30,37)(21,33,36,46)(25,40,34,48)(26,32,27,43)(29,45,47,35)(38,41,44,42);(17,20,22,31,23,30,37,19)(18,24,27,32,28,39,26,43)(21,33,35,47,36,46,45,29)(25,38,42,40,34,44,41,48);(2,7)(6,9)(8,15)(14,16);(1,4,5,13)(2,8,7,15)(3,10,11,12)(6,14,9,16)(17,21,22,35,23,36,37,45)(18,25,27,42,28,34,26,41)(19,29,20,33,31,47,30,46)(24,38,32,40,39,44,43,48)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6716869);