#411265

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×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

920 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)(5,6,12,28,10,16,11,17)(7,13,19,22,20,31,27,34)(8,14,29,36,23,32,24,33)(9,21,25,35,26,15,30,18);(1,2,4,3)(5,7,11,27,10,20,12,19)(6,13,28,22,16,31,17,34)(8,21,24,18,23,15,29,35)(9,14,25,36,26,32,30,33);(1,2,4,3)(5,8,11,24,10,23,12,29)(6,14,17,33,16,32,28,36)(7,18,19,21,20,35,27,15)(9,22,25,31,26,34,30,13);(1,3,4,2)(5,9)(6,15)(7,14)(8,22)(10,26)(11,25)(12,30)(13,29)(16,21)(17,18)(19,33)(20,32)(23,34)(24,31)(27,36)(28,35)

Polycyclic presentation:

2^9|a^2=efg,b^2=gh,c^2=fg,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, 318461);