#6432418

order	512
nilpotency class	3
isoclinism	isoclinic class of #6431707
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3166 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339614);