#508963

order	512
nilpotency class	4
isoclinism	isoclinic class of #508292
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

530 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343806, #72343823, #72343831
maximal normal subgroup of	#72343806, #72343823, #72343831

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416159);