#495165

order	512
nilpotency class	4
isoclinism	isoclinic class of #480460
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂×#534	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄ ◁ C₂×#518 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1343 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(2,4)(5,6,9,14,10,13,22,23)(7,15,18,29,17,11,21,25)(8,12,20,27,19,26,32,34)(16,28,31,36,30,24,33,35);(5,7,9,21)(6,11,23,29)(8,16,20,33)(10,17,22,18)(12,24,34,36)(13,15,14,25)(19,30,32,31)(26,28,27,35);(1,2,3,4)(5,8)(6,12)(7,16)(9,20)(10,19)(11,24)(13,26)(14,27)(15,28)(17,30)(18,31)(21,33)(22,32)(23,34)(25,35)(29,36);(8,19)(12,26)(16,30)(20,32)(24,28)(27,34)(31,33)(35,36)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=f,g^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=eh,g^b=gh,h^b=hi,d^c=di,g^e=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402361);