#416569

order	512
nilpotency class	3
isoclinism	isoclinic class of #394801
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1696 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=eh,c^2=f,d^2=h,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 323765);