#360126

order	512
nilpotency class	3
isoclinism	isoclinic class of #196122
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1579 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=i,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,g^b=gi,g^c=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, 267322);