#397946

order	512
nilpotency class	3
isoclinism	isoclinic class of #394823
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1864 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:

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

Polycyclic presentation:

2^9|b^2=ei,c^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^c=ei,g^c=gi,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 305142);