#9054395

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

5155 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=gh,d^2=g,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8961591);