#6814481

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806671
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	#42806	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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=gh,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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6721677);