#8466917

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

4929 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)(9,10)(11,14)(12,15)(13,16)(17,18,23,22)(19,24,29,27)(20,25,30,32)(21,31,28,26);(1,3)(2,5)(4,8)(6,7)(9,11,13,15)(10,14,16,12)(17,18,23,22)(19,27,29,24)(20,25,30,32)(21,26,28,31);(9,11)(10,12)(13,15)(14,16)(17,19,23,29)(18,24,22,27)(20,28,30,21)(25,26,32,31);(2,6)(5,7)(9,11)(10,14)(12,16)(13,15)(17,20)(18,25)(19,28)(21,29)(22,32)(23,30)(24,26)(27,31);(1,3)(2,5)(4,8)(6,7)(17,21,23,28)(18,26,22,31)(19,20,29,30)(24,32,27,25);(1,2)(3,5)(4,6)(7,8)(9,12,13,14)(10,15,16,11)(17,22,23,18)(19,27,29,24)(20,25,30,32)(21,31,28,26)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=hi,e^2=hi,f^2=i,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, 8374113);