#6453766

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437418
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3455	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3155 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)(16,28)(17,22)(18,32)(19,23)(20,26)(21,30)(25,39)(27,35)(29,43)(31,37)(33,36)(34,44)(38,41)(40,42);(1,2)(3,4)(13,15,19,27)(14,20,23,36)(16,25,30,41)(17,26,18,33)(21,34,28,42)(22,35,32,24)(29,40,31,44)(37,39,43,38);(2,4)(6,9)(10,12)(13,16,19,30)(14,21,23,28)(15,25,27,41)(17,29,18,31)(20,34,36,42)(22,37,32,43)(24,38,35,39)(26,40,33,44);(2,4)(13,17,19,18)(14,22,23,32)(15,26,27,33)(16,29,30,31)(20,35,36,24)(21,37,28,43)(25,40,41,44)(34,39,42,38);(5,7,8,11)(6,10,9,12)(16,30)(21,28)(25,41)(29,31)(34,42)(37,43)(38,39)(40,44)

Polycyclic presentation:

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

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, 6360962);