#8934540

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

4996 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)(11,15,16,13)(17,18,22,25)(19,26,27,23)(20,30,24,21)(28,32,31,29);(1,3,5,8)(2,6,7,4)(9,10)(11,13)(12,14)(15,16)(19,27)(23,26)(28,31)(29,32);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(17,19,22,27)(18,23,25,26)(20,28,24,31)(21,32,30,29);(2,7)(3,8)(9,10,12,14)(11,13,16,15)(17,19)(18,23)(20,31)(21,29)(22,27)(24,28)(25,26)(30,32);(1,3)(2,4)(5,8)(6,7)(9,10)(11,13)(12,14)(15,16)(17,20,22,24)(18,21,25,30)(19,28,27,31)(23,32,26,29);(1,4)(2,3)(5,6)(7,8)(9,11,12,16)(10,13,14,15)(17,21)(18,24)(19,29)(20,25)(22,30)(23,28)(26,31)(27,32)

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=gh,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, 8841736);