#9040945

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

4780 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)(3,4)(5,8)(6,7)(9,10)(11,15)(12,13)(14,16)(17,18,24,22)(19,28,27,23)(20,26,21,25)(29,32,31,30);(1,3)(2,6)(4,8)(5,7)(10,13)(14,15)(17,19)(18,23)(20,29)(21,31)(22,28)(24,27)(25,32)(26,30);(1,2)(3,6)(4,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,20,24,21)(18,25,22,26)(19,29,27,31)(23,32,28,30);(1,4,5,6)(2,7,8,3)(9,11,12,16)(10,14,13,15)(17,21,24,20)(18,26,22,25)(19,29,27,31)(23,32,28,30);(1,4)(2,3)(5,6)(7,8)(17,22,24,18)(19,28,27,23)(20,25,21,26)(29,32,31,30);(1,3)(2,4)(5,7)(6,8)(17,23,24,28)(18,27,22,19)(20,30,21,32)(25,29,26,31)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8948141);