#6810446

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806671
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#42995	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1996 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)(17,31)(21,37)(27,41)(30,38)(32,36)(33,42)(39,44)(40,43);(5,7,8,12)(6,9,10,11)(13,14,18,23)(15,24,29,35)(16,20,25,22)(17,32,31,36)(19,26,34,28)(21,38,37,30)(27,42,41,33)(39,40,44,43);(2,4)(7,12)(9,11)(13,15,25,26,18,29,16,28)(14,19,20,35,23,34,22,24)(17,27,38,40,31,41,30,43)(21,33,36,44,37,42,32,39);(6,10)(9,11)(13,16,18,25)(14,20,23,22)(15,26,29,28)(17,30,31,38)(19,24,34,35)(21,36,37,32)(27,40,41,43)(33,39,42,44);(13,17,25,38,18,31,16,30)(14,21,20,36,23,37,22,32)(15,27,26,40,29,41,28,43)(19,33,35,44,34,42,24,39)

Polycyclic presentation:

2^9|b^2=fi,c^2=fhi,d^2=i,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, 6717642);