#499441

order	512
nilpotency class	4
isoclinism	isoclinic class of #498966
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	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

584 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=e,c^2=e,d^2=i,e^2=i,f^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,d^b=di,f^c=fh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320223, #72320244, #72320401, …
maximal normal subgroup of	#72320223, #72320244, #72320401, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406637);