#501691

order	512
nilpotency class	4
isoclinism	isoclinic class of #500854
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#24797	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

616 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)(5,6,10,15)(7,18,22,27)(8,24,26,23)(9,28,30,33)(11,16,29,17)(12,13,31,21)(14,25,36,19)(20,35,32,34);(5,7,19,34,11,24,30,13,10,22,25,35,29,23,9,21)(6,12,28,8,17,32,36,27,15,31,33,26,16,20,14,18);(2,4)(5,6)(7,20)(8,13)(9,14)(10,15)(11,17)(12,24)(16,29)(18,34)(19,28)(21,26)(22,32)(23,31)(25,33)(27,35)(30,36);(5,8,25,20,11,27,9,12,10,26,19,32,29,18,30,31)(6,13,33,7,17,35,14,24,15,21,28,22,16,34,36,23)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408887);