#6630244

order	512
nilpotency class	3
isoclinism	isoclinic class of #6628851
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	C₂×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2191 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)(7,10)(8,9)(11,12)(13,14,17,22)(15,24,26,19)(16,20,28,35)(18,21,30,23)(25,37,38,31)(27,33,40,32)(29,34,41,36)(39,43,44,42);(6,9)(10,12)(13,15,17,26)(14,19,22,24)(16,25,28,38)(18,27,30,40)(20,31,35,37)(21,32,23,33)(29,39,41,44)(34,42,36,43);(1,2,3,4)(13,16)(14,20)(15,25)(17,28)(18,29)(19,31)(21,34)(22,35)(23,36)(24,37)(26,38)(27,39)(30,41)(32,42)(33,43)(40,44);(14,21)(18,30)(19,32)(20,34)(22,23)(24,33)(27,40)(29,41)(31,42)(35,36)(37,43)(39,44);(5,7,8,11)(6,10,9,12)(16,28)(20,35)(25,38)(29,41)(31,37)(34,36)(39,44)(42,43)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6537440);