#6398487

order	512
nilpotency class	2
isoclinism	isoclinic class of #45438
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	#52249	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1518 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=fh,d^2=f,e^2=g,h^2=i,b^a=bf,c^a=cg,e^a=efi,d^b=dfi,d^c=di,e^d=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6305683);