#6806701

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	#42643	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2015 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)(13,14)(15,20)(16,27)(17,32)(18,23)(19,24)(21,29)(22,33)(25,41)(26,36)(28,40)(30,37)(31,44)(34,39)(35,43)(38,42);(5,7)(6,9)(8,12)(10,11)(13,15,18,26)(14,20,23,36)(16,28,29,30)(17,19,33,34)(21,37,27,40)(22,39,32,24)(25,42,43,31)(35,44,41,38);(2,4)(13,16,19,31,18,29,34,42)(14,21,24,38,23,27,39,44)(15,25,17,28,26,43,33,30)(20,35,32,37,36,41,22,40);(6,10)(9,11)(16,29)(21,27)(25,43)(28,30)(31,42)(35,41)(37,40)(38,44);(7,12)(9,11)(13,17,19,26,18,33,34,15)(14,22,24,20,23,32,39,36)(16,30,31,25,29,28,42,43)(21,37,38,41,27,40,44,35)

Polycyclic presentation:

2^9|b^2=i,c^2=h,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, 6713897);