#365074

order	512
nilpotency class	3
isoclinism	isoclinic class of #220200
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3060	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1618 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 272270);