#8934408

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

4900 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,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, 8841604);