#9947496

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

5092 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:

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

Polycyclic presentation:

2^9|b^2=i,d^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,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, 9854692);