#6466026

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438955
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2016 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=ghi,d^2=h,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi,e^c=ei,e^d=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, 6373222);