#401796

order	512
nilpotency class	3
isoclinism	isoclinic class of #396113
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1175 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,g^b=gi,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, 308992);