#401795

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

Polycyclic presentation:

2^9|a^2=g,c^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, 308991);