#6814897

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1880 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)(13,14,20,26)(15,21,28,36)(16,22,33,41)(17,23,19,25)(18,24,34,42)(27,35,43,44)(29,37,31,39)(30,38,32,40);(5,7)(6,9)(8,12)(10,11)(13,15,20,28)(14,21,26,36)(16,29,33,31)(17,18,19,34)(22,37,41,39)(23,24,25,42)(27,30,43,32)(35,38,44,40);(1,2,3,4)(13,16,19,32,20,33,17,30)(14,22,25,40,26,41,23,38)(15,27,18,29,28,43,34,31)(21,35,24,37,36,44,42,39);(6,10)(9,11)(13,17,20,19)(14,23,26,25)(15,18,28,34)(16,30,33,32)(21,24,36,42)(22,38,41,40)(27,29,43,31)(35,37,44,39);(7,12)(9,11)(13,18,19,28,20,34,17,15)(14,24,25,36,26,42,23,21)(16,31,32,27,33,29,30,43)(22,39,40,35,41,37,38,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gh,d^2=i,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,e^d=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, 6722093);