#6815878

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806671
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	#42883	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2131 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:

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

Polycyclic presentation:

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

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, 6723074);