#6629725

order	512
nilpotency class	3
isoclinism	isoclinic class of #32932
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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
prime radical	C₂×#3054	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2308 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:

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

Polycyclic presentation:

2^9|b^2=fg,d^2=f,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,h^a=hi,c^b=ci

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6536921);