#8437832

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4772 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=hi,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8345028);