#181246

order	512
nilpotency class	3
isoclinism	isoclinic class of #181129
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

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

1653 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)(3,11)(4,8)(5,9)(6,10)(7,12)(13,16)(14,15)(17,18)(19,25,27,22)(20,23)(21,24)(26,31,32,29)(28,30);(1,3,5,13,6,14,4,12)(2,7,8,15,10,16,9,11)(17,19)(18,22)(20,26)(21,27)(23,29)(24,25)(28,32)(30,31);(1,4,6,5)(2,8,10,9)(3,12,14,13)(7,15,16,11)(17,20,21,28)(18,23,24,30)(19,26,27,32)(22,29,25,31);(20,28)(23,30)(26,32)(29,31)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88442);