#6494233

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470003
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	#2619	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1824 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)(7,11)(10,12)(13,14,17,23,18,22,32,39)(15,24,26,35,27,40,31,19)(16,21,30,38,29,20,36,37)(25,42,43,33,28,41,44,34);(5,6)(7,10)(8,9)(11,12)(13,15,18,27)(14,19,22,35)(16,28,29,25)(17,31,32,26)(20,34,21,33)(23,40,39,24)(30,43,36,44)(37,41,38,42);(1,2,3,4)(6,9)(10,12)(13,16)(14,20)(15,25)(17,30)(18,29)(19,33)(21,22)(23,37)(24,41)(26,43)(27,28)(31,44)(32,36)(34,35)(38,39)(40,42);(6,9)(10,12)(13,16)(14,21)(15,25)(17,30)(18,29)(19,34)(20,22)(23,38)(24,42)(26,43)(27,28)(31,44)(32,36)(33,35)(37,39)(40,41);(5,7,8,11)(6,10,9,12)(14,22)(16,29)(19,35)(23,39)(24,40)(25,28)(30,36)(43,44)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6401429);