#6814485

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	#42674	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1852 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)(7,11)(8,10)(9,12)(13,14)(15,20)(16,28)(17,33)(18,23)(19,24)(21,30)(22,27)(25,40)(26,36)(29,43)(31,37)(32,44)(34,39)(35,42)(38,41);(5,7)(6,9)(8,12)(10,11)(13,15,18,26)(14,20,23,36)(16,29,30,31)(17,34,27,19)(21,37,28,43)(22,24,33,39)(25,41,42,32)(35,44,40,38);(1,2,3,4)(13,16,19,32,18,30,34,41)(14,21,24,38,23,28,39,44)(15,25,27,29,26,42,17,31)(20,35,22,37,36,40,33,43);(6,10)(9,11)(16,30)(21,28)(25,42)(29,31)(32,41)(35,40)(37,43)(38,44);(7,12)(9,11)(13,17,19,15,18,27,34,26)(14,22,24,36,23,33,39,20)(16,31,32,25,30,29,41,42)(21,37,38,40,28,43,44,35)

Polycyclic presentation:

2^9|b^2=i,c^2=ghi,e^2=h,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, 6721681);