#6467112

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	#35210	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2002 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)(7,10)(8,9)(11,12)(13,14)(15,20)(16,21)(17,31)(18,24)(19,23)(22,32)(25,35)(26,44)(27,38)(28,37)(29,36)(30,39)(33,42)(34,41)(40,43);(2,4)(13,15)(14,20)(16,29)(17,25)(18,33)(19,28)(21,36)(22,40)(23,37)(24,42)(26,30)(27,34)(31,35)(32,43)(38,41)(39,44);(6,9)(10,12)(13,16,18,17,19,30,34,32)(14,21,24,31,23,39,41,22)(15,25,27,29,28,43,33,26)(20,35,38,36,37,40,42,44);(13,17,18,30,19,32,34,16)(14,22,24,21,23,31,41,39)(15,26,27,25,28,29,33,43)(20,36,38,40,37,44,42,35);(5,7,8,11)(6,10,9,12)(14,23)(16,30)(17,32)(20,37)(24,41)(25,43)(26,29)(38,42)

Polycyclic presentation:

2^9|c^2=h,d^2=h,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,d^b=dh,h^b=hi,d^c=di,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, 6374308);