#361070

order	512
nilpotency class	3
isoclinism	isoclinic class of #188770
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

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	#2283
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #1729 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

877 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)(7,16,19,11)(8,20,22,13)(9,15)(10,14)(12,18,26,32)(17,24)(21,34,35,27)(23,28)(25,30)(29,31)(33,36);(1,2)(3,4)(5,7)(6,11)(8,17)(9,18)(10,19)(12,15)(13,24)(14,16)(20,29)(21,30)(22,31)(23,32)(25,27)(26,28)(33,34)(35,36);(2,4)(5,7)(6,12,14,26)(8,17)(9,18)(10,19)(11,15,16,28)(13,25,20,33)(21,30)(22,31)(23,32)(24,27,29,34)(35,36);(5,8,9,21,10,22,23,35)(6,13,15,27,14,20,28,34)(7,17,18,30,19,31,32,36)(11,24,12,25,16,29,26,33)

Polycyclic presentation:

2^9|a^2=h,c^2=e,d^2=f,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,h^b=hi,h^c=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, 268266);