#181131

order	512
nilpotency class	3
isoclinism	isoclinic class of #181131
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#95
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1586 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=hi,b^a=be,e^a=eh,g^a=ghi,c^b=ch,f^b=fi,d^c=di

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, 88327);