#364931

order	512
nilpotency class	3
isoclinism	isoclinic class of #196290
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	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1766 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,f^c=fi,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, 272127);