#394083

order	512
nilpotency class	3
isoclinism	isoclinic class of #393982
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3968 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)(9,20)(12,22)(13,23)(14,21)(15,27)(16,29)(18,31)(24,34)(25,30)(26,32)(33,35);(1,2)(3,4)(5,6,10,14)(7,12,17,25)(8,13,19,26)(9,21,20,11)(15,24,27,35)(16,30,29,22)(18,32,31,23)(28,34,36,33);(2,4)(5,7)(6,12)(8,15)(9,16)(10,17)(11,22)(13,24)(14,25)(18,28)(19,27)(20,29)(21,30)(23,33)(26,35)(31,36)(32,34);(5,8)(6,13)(7,15)(9,18)(10,19)(11,23)(12,24)(14,26)(16,28)(17,27)(20,31)(21,32)(22,33)(25,35)(29,36)(30,34)

Polycyclic presentation:

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

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, 301279);