#9778489

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

5055 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9685685);