#504126

order	512
nilpotency class	4
isoclinism	isoclinic class of #504125
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#476
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3050	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

538 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)(3,4)(5,6,10,15,12,16,9,11)(7,13,19,30,21,20,18,28)(8,22,24,25,27,31,26,14)(17,23,32,35,33,36,34,29);(2,4)(5,7,10,19,12,21,9,18)(6,13,15,30,16,20,11,28)(8,23,24,35,27,36,26,29)(14,17,22,32,25,33,31,34);(5,8)(6,14)(7,17,21,33)(9,24)(10,26)(11,22)(12,27)(13,29,20,35)(15,31)(16,25)(18,32,19,34)(23,30,36,28);(2,4)(5,9,12,10)(6,11,16,15)(7,18,21,19)(8,24,27,26)(13,28,20,30)(14,22,25,31)(17,32,33,34)(23,35,36,29)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=fh,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=gh,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322971
maximal normal subgroup of	#72322971

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411322);