#6809877

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806623
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2059 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=fh,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6717073);