#393644

order	512
nilpotency class	3
isoclinism	isoclinic class of #32935
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×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1312 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)(13,14,19,24,20,25,18,36)(15,26,30,39,31,42,29,21)(16,32,34,41,35,23,17,22)(27,40,28,44,33,37,43,38);(1,2,3,4)(13,15)(14,21,25,39)(16,33)(17,28)(18,29)(19,30)(20,31)(22,40,41,37)(23,38,32,44)(24,26,36,42)(27,35)(34,43);(6,9)(10,12)(13,16,18,17,20,35,19,34)(14,22,24,32,25,41,36,23)(15,27,29,43,31,33,30,28)(21,37,26,38,39,40,42,44);(5,7,8,11)(6,10,9,12)(13,17,18,35,20,34,19,16)(14,23,24,22,25,32,36,41)(15,28,29,27,31,43,30,33)(21,38,26,40,39,44,42,37)

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,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, 300840);