#405486

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	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1420 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)(5,6,11,15)(7,16,19,33)(8,20,24,12)(9,13,26,18)(10,27,28,14)(17,32,35,29)(21,36,34,22)(23,30,25,31);(1,2,4,3)(5,7,10,18,11,19,28,13)(6,9,14,16,15,26,27,33)(8,21,25,35,24,34,23,17)(12,29,31,36,20,32,30,22);(1,3,4,2)(5,8,10,23,11,24,28,25)(6,12,14,30,15,20,27,31)(7,17,18,34,19,35,13,21)(9,22,16,32,26,36,33,29);(5,9)(6,13)(7,14)(8,22)(10,16)(11,26)(12,21)(15,18)(17,30)(19,27)(20,34)(23,32)(24,36)(25,29)(28,33)(31,35)

Polycyclic presentation:

2^9|a^2=i,b^2=eh,c^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,f^c=fi,g^c=gi,h^c=hi,f^d=fi

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, 312682);