#9847927

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

5142 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=ghi,d^2=g,e^2=i,f^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, 9755123);