#512305

order	512
nilpotency class	4
isoclinism	isoclinic class of #511893
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#469	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

610 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)(6,11)(7,15)(8,18)(9,20)(12,27)(13,21)(14,26)(16,35)(17,34)(19,23)(22,31)(24,30)(28,33)(29,32);(1,2,4,3)(5,6,8,11,10,14,18,26)(7,12,19,24,17,29,16,28)(9,21,20,36,22,13,31,25)(15,33,35,32,34,30,23,27);(9,22)(13,21)(15,34)(20,31)(23,35)(25,36)(27,32)(30,33);(1,3,4,2)(5,7,10,17)(6,12,14,29)(8,19,18,16)(9,23,22,35)(11,24,26,28)(13,32,21,27)(15,31,34,20)(25,30,36,33)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419501);