#493038

order	512
nilpotency class	4
isoclinism	isoclinic class of #33359
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	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄ ◁ C₂×#519 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1274 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)(8,19)(9,14)(10,12)(11,16)(13,21)(17,33)(18,29)(20,28)(22,27)(23,26)(24,32)(25,34)(30,35)(31,36);(5,7,9,17,10,18,23,35)(6,11,14,25,12,24,26,36)(8,16,20,31,21,32,22,34)(13,15,27,30,19,29,28,33);(1,3,4,2)(5,6)(7,11)(8,13)(9,14)(10,12)(15,16)(17,25)(18,24)(19,21)(20,27)(22,28)(23,26)(29,32)(30,31)(33,34)(35,36);(6,12)(11,24)(13,19)(14,26)(15,29)(25,36)(27,28)(30,33)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,e^2=i,g^2=i,b^a=be,d^a=di,e^a=ei,g^a=ghi,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, 400234);