#502088

order	512
nilpotency class	4
isoclinism	isoclinic class of #502081
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

805 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=i,d^2=i,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,h^b=hi,f^c=fi,g^c=ghi,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72323026
maximal normal subgroup of	#72323026

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409284);