#493299

order	512
nilpotency class	2
isoclinism	isoclinic class of #534
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₈	C₂⁴
commutator	C₂²	C₂²×C₄×C₈
socle	C₂³	C₂³×C₈
prime radical	C₂⁴	C₂²×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

632 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=e,f^2=h,g^2=h,h^2=i,b^a=be,d^a=de,c^b=cei,d^c=di

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400495);