#181346

order	512
nilpotency class	3
isoclinism	isoclinic class of #181344
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	C₂³	C₂²×C₄²
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2030 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,b^a=be,e^a=eh,g^a=gi,e^b=ei,f^b=fh,d^c=dh

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, 88542);