#502200

order	512
nilpotency class	4
isoclinism	isoclinic class of #500733
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	#33526	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

746 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,d^2=i,e^2=hi,f^2=i,g^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,e^b=eh,h^b=hi,f^c=fi,g^c=gi,e^d=eh,h^d=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409396);