#501700

order	512
nilpotency class	4
isoclinism	isoclinic class of #501698
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#24803	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ 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:

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

Polycyclic presentation:

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

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, 408896);