#181257

order	512
nilpotency class	3
isoclinism	isoclinic class of #181134
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1458 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:

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

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=dhi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88453);