#394293

order	512
nilpotency class	3
isoclinism	isoclinic class of #394292
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2618 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:

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

Polycyclic presentation:

2^9|b^2=ei,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,f^c=fi,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, 301489);