#6808004

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	#42880	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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=hi,d^2=f,e^2=hi,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, 6715200);