#505733

order	512
nilpotency class	4
isoclinism	isoclinic class of #500829
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

599 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)(11,13)(12,14)(15,16)(17,18,21,27,32,30,31,25,23,28,20,26,22,19,24,29);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19,20,25,32,18,24,26,23,30,21,29,22,28,31,27);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(18,25)(19,26)(20,31)(21,24)(22,32)(27,30)(28,29);(2,5)(6,8)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343663, #72343833, #72343855
maximal normal subgroup of	#72343663, #72343833, #72343855

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 412929);