#393582

order	512
nilpotency class	3
isoclinism	isoclinic class of #32935
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

922 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=f,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,g^a=gi,d^c=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 300778);