#360903

order	512
nilpotency class	3
isoclinism	isoclinic class of #188770
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×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	#2283
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #1729 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1021 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,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, 268099);