#360164

order	512
nilpotency class	3
isoclinism	isoclinic class of #196278
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

1627 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)(5,6,10,16,11,15,9,14)(7,12,18,26,19,27,24,31)(8,20,22,13,23,30,21,29)(17,32,33,28,34,36,35,25);(1,2,4,3)(5,7)(6,12)(8,17)(9,24)(10,18)(11,19)(13,28)(14,31)(15,27)(16,26)(20,32)(21,35)(22,33)(23,34)(25,29)(30,36);(5,8,11,23)(6,13,15,29)(7,17,19,34)(9,22,10,21)(12,25,27,28)(14,30,16,20)(18,35,24,33)(26,36,31,32);(1,3,4,2)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=i,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,f^c=fi,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, 267360);