#362840

order	512
nilpotency class	3
isoclinism	isoclinic class of #24390
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

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	C₂²×C₂²⋊C₄
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1414 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=eg,f^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di,f^c=fi,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 270036);