#393983

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

3946 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:

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

Polycyclic presentation:

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

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, 301179);