#493031

order	512
nilpotency class	4
isoclinism	isoclinic class of #33348
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×#99
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄ ◁ C₂×#518 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1340 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:

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

Polycyclic presentation:

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

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, 400227);