#502082

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,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, 409278);