#504435

order	512
nilpotency class	4
isoclinism	isoclinic class of #504434
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#26601
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1773 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

576 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411631);