#365073

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×#3058	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1756 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)(3,4)(5,6)(7,15)(8,12,22,26)(9,17)(10,18)(11,14)(13,23)(16,21,19,28)(20,30,34,32)(24,31)(25,35,36,33)(27,29);(5,7,15,6,11,18,10,14)(8,19,21,12,22,16,28,26)(9,17,29,13,24,31,23,27)(20,32,33,25,34,30,35,36);(2,4)(5,8)(6,12)(7,16)(9,20)(10,21)(11,22)(13,25)(14,26)(15,28)(17,30)(18,19)(23,33)(24,34)(27,36)(29,35)(31,32);(5,9,15,29,11,24,10,23)(6,13,18,31,14,27,7,17)(8,20,28,35,22,34,21,33)(12,25,19,32,26,36,16,30)

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,d^2=ei,e^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, 272269);