#8934428

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

5002 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)(17,18,23,25)(19,27,30,24)(20,21,29,31)(22,32,28,26);(1,3,5,8)(2,6,7,4)(11,15)(13,16)(17,19,23,30)(18,24,25,27)(20,22,29,28)(21,26,31,32);(1,3)(2,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(18,25)(20,29)(22,28)(24,27);(2,7)(3,8)(17,20,23,29)(18,21,25,31)(19,28,30,22)(24,32,27,26);(1,3)(2,4)(5,8)(6,7)(9,11,12,15)(10,13,14,16)(17,21)(18,20)(19,26)(22,24)(23,31)(25,29)(27,28)(30,32);(1,4)(2,3)(5,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,22,23,28)(18,26,25,32)(19,29,30,20)(21,24,31,27)

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,d^2=g,e^2=gh,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, 8841624);