#393238

order	512
nilpotency class	3
isoclinism	isoclinic class of #32935
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×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	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1212 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=f,d^2=e,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, 300434);