#6447640

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437753
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

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

3082 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6354836);