#8463693

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

5356 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)(17,18,22,24)(19,25,27,21)(20,23,29,30)(26,31,32,28);(1,3)(2,5)(4,8)(6,7)(11,16)(13,14)(17,19,22,27)(18,21,24,25)(20,26,29,32)(23,28,30,31);(9,11)(10,13)(12,16)(14,15)(17,18)(19,21)(20,23)(22,24)(25,27)(26,28)(29,30)(31,32);(2,6)(5,7)(9,11)(10,14)(12,16)(13,15)(17,18)(19,25)(20,23)(21,27)(22,24)(26,31)(28,32)(29,30);(1,3)(2,5)(4,8)(6,7)(17,20,22,29)(18,23,24,30)(19,26,27,32)(21,28,25,31);(1,2)(3,5)(4,6)(7,8)(9,10)(11,13)(12,15)(14,16)(17,21,22,25)(18,19,24,27)(20,28,29,31)(23,26,30,32)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,e^2=hi,f^2=hi,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, 8370889);