#9848528

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

4703 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=i,d^2=g,e^2=ghi,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, 9755724);