#493298

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄²
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

642 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)(13,14,17,23,19,15,21,27,20,25,18,24,34,29,35,28)(16,22,31,38,33,26,36,41,30,37,32,39,43,40,44,42);(6,9)(7,11)(13,15,18,28,19,25,35,23,20,29,17,27,34,14,21,24)(16,26,32,42,33,37,44,38,30,40,31,41,43,22,36,39);(5,7,8,11)(6,10,9,12)(13,16)(14,22)(15,26)(17,31)(18,32)(19,33)(20,30)(21,36)(23,38)(24,39)(25,37)(27,41)(28,42)(29,40)(34,43)(35,44);(6,9)(10,12)(16,30)(22,37)(26,40)(31,32)(33,43)(36,44)(38,39)(41,42)

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=ce,d^c=di

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400494);