#393649

order	512
nilpotency class	3
isoclinism	isoclinic class of #32941
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄³

Aliases

No aliases for this group.

Subgroups

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

1292 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)(7,10)(8,9)(11,12)(13,14,18,25,19,23,32,24)(15,20,30,37,28,38,29,40)(16,31,17,21,33,22,34,39)(26,41,43,44,42,36,27,35);(1,2,3,4)(13,15,18,29,19,28,32,30)(14,20,24,37,23,38,25,40)(16,26,17,27,33,42,34,43)(21,35,31,41,39,44,22,36);(6,9)(10,12)(13,16,32,34,19,33,18,17)(14,21,25,22,23,39,24,31)(15,26,30,43,28,42,29,27)(20,35,40,36,38,44,37,41);(5,7,8,11)(6,10,9,12)(13,17,18,33,19,34,32,16)(14,22,24,21,23,31,25,39)(15,27,29,42,28,43,30,26)(20,36,37,35,38,41,40,44)

Polycyclic presentation:

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

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, 300845);