#499506

order	512
nilpotency class	4
isoclinism	isoclinic class of #13309
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#469
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#23697
prime radical	#13309	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

591 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72217287, #72217290, #72224959, …
maximal normal subgroup of	#72217287, #72217290, #72224959, …
maximal quotient of	C₂×#499506

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406702);