#411267

order	512
nilpotency class	3
isoclinism	isoclinic class of #399126
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2504
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

960 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:

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

Polycyclic presentation:

2^9|a^2=efgi,b^2=gh,c^2=fgi,d^2=f,g^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,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, 318463);