#6454248

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437537
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	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1994 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)(7,10)(8,9)(11,12)(13,14)(15,23)(16,21)(17,20)(18,22)(19,26)(24,33)(25,41)(27,39)(28,44)(29,35)(30,38)(31,36)(32,40)(34,42)(37,43);(2,4)(13,15)(14,19)(16,28)(17,26)(18,24)(20,23)(21,37)(22,39)(25,30)(27,31)(29,43)(32,42)(33,36)(34,38)(35,44)(40,41);(6,9)(10,12)(14,20)(15,24)(18,31)(19,33)(21,35)(23,39)(25,28)(26,27)(30,32)(34,44)(37,41)(42,43);(13,16,17,29)(14,21,20,35)(15,25,26,42)(18,32,31,30)(19,34,23,41)(22,40,36,38)(24,28,27,43)(33,44,39,37);(5,7,8,11)(6,10,9,12)(14,20)(15,26)(21,35)(22,36)(24,27)(25,42)(28,43)(38,40)

Polycyclic presentation:

2^9|a^2=fi,d^2=i,e^2=gi,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,e^b=ei,h^b=hi,h^c=hi,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6361444);