#509340

order	512
nilpotency class	4
isoclinism	isoclinic class of #509340
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#30593
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

557 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416536);