#360525

order	512
nilpotency class	3
isoclinism	isoclinic class of #188770
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#380
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#407 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2186 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 267721);