#9863072

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

4654 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=h,d^2=i,e^2=ghi,f^2=h,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, 9770268);