#6448693

order	512
nilpotency class	3
isoclinism	isoclinic class of #23924
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3318 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6355889);