#502192

order	512
nilpotency class	4
isoclinism	isoclinic class of #499769
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	C₂×#2192	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

753 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=i,d^2=i,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,e^b=eh,g^b=gi,h^b=hi,g^c=gi,e^d=eh,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, 409388);