#410849

order	512
nilpotency class	3
isoclinism	isoclinic class of #399126
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2504
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1221 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=g,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,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, 318045);