#6810098

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	#42796	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2065 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)(16,27)(20,36)(25,29)(28,42)(31,37)(35,38)(40,43)(41,44);(5,7)(6,9)(8,12)(10,11)(13,14,18,22)(15,23,24,39)(16,28,27,42)(17,32,30,21)(19,33,34,26)(20,37,36,31)(25,38,29,35)(40,44,43,41);(2,4)(7,12)(9,11)(13,15,17,26,18,24,30,33)(14,19,21,23,22,34,32,39)(16,29,31,44,27,25,37,41)(20,38,28,43,36,35,42,40);(6,10)(9,11)(15,24)(19,34)(23,39)(25,29)(26,33)(35,38)(40,43)(41,44);(13,16,30,37,18,27,17,31)(14,20,32,42,22,36,21,28)(15,25,33,44,24,29,26,41)(19,35,39,43,34,38,23,40)

Polycyclic presentation:

2^9|b^2=i,c^2=fh,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,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, 6717294);