#393641

order	512
nilpotency class	3
isoclinism	isoclinic class of #32932
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄³

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	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1304 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)(13,14,19,24,20,26,18,25)(15,27,31,39,32,38,30,21)(16,23,17,22,35,41,34,42)(28,36,43,37,33,40,29,44);(1,2,3,4)(13,15)(14,21,26,39)(16,33)(17,29)(18,30)(19,31)(20,32)(22,40,42,36)(23,37,41,44)(24,27,25,38)(28,35)(34,43);(6,9)(10,12)(13,16,18,34,20,35,19,17)(14,22,24,41,26,42,25,23)(15,28,30,29,32,33,31,43)(21,36,27,37,39,40,38,44);(5,7,8,11)(6,10,9,12)(13,17,19,35,20,34,18,16)(14,23,25,42,26,41,24,22)(15,29,31,28,32,43,30,33)(21,37,38,36,39,44,27,40)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=g,d^2=f,e^2=i,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,c^b=ci

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, 300837);