#360524

order	512
nilpotency class	3
isoclinism	isoclinic class of #188770
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

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

2246 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)(7,15,18,11)(8,19)(9,22)(10,13)(12,20)(14,23)(16,29,31,33)(17,25,32,28)(21,34)(24,30,27,36)(26,35);(1,2)(3,4)(5,7)(6,11)(8,16)(9,17)(10,18)(12,24)(13,15)(14,25)(19,33)(20,30)(21,31)(22,28)(23,32)(26,27)(29,34)(35,36);(2,4)(5,8,9,20,10,21,23,35)(6,12,14,19,13,26,22,34)(7,16,17,30,18,31,32,36)(11,24,25,33,15,27,28,29);(5,7)(6,11)(8,16)(9,17)(10,18)(12,24)(13,15)(14,25)(19,33)(20,30)(21,31)(22,28)(23,32)(26,27)(29,34)(35,36)

Polycyclic presentation:

2^9|a^2=h,c^2=f,f^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, 267720);