#6808013

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
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	#42800	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1826 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)(6,9)(7,11)(17,30)(22,38)(26,32)(29,33)(31,43)(36,39)(40,44)(41,42);(5,6)(7,10)(8,9)(11,12)(13,14,19,23)(15,24,27,35)(16,21,18,34)(17,31,30,43)(20,28,37,25)(22,29,38,33)(26,39,32,36)(40,42,44,41);(2,4)(13,15,16,28,19,27,18,25)(14,20,34,35,23,37,21,24)(17,32,29,41,30,26,33,42)(22,39,43,40,38,36,31,44);(5,7,8,11)(6,10,9,12)(13,16,19,18)(14,21,23,34)(15,25,27,28)(17,29,30,33)(20,35,37,24)(22,31,38,43)(26,41,32,42)(36,44,39,40);(6,9)(10,12)(13,17,18,33,19,30,16,29)(14,22,21,31,23,38,34,43)(15,26,25,41,27,32,28,42)(20,36,24,40,37,39,35,44)

Polycyclic presentation:

2^9|b^2=i,c^2=hi,d^2=fi,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,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6715209);