#8934536

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

5026 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)(11,13)(12,15)(14,16)(17,18,23,27)(19,28,25,21)(20,24,30,32)(22,26,29,31);(1,3,5,8)(2,6,7,4)(9,10,14,16)(11,13,15,12)(17,19,23,25)(18,21,27,28)(20,22,30,29)(24,31,32,26);(1,3)(2,6)(4,7)(5,8)(9,11,14,15)(10,12,16,13)(20,30)(22,29)(24,32)(26,31);(2,7)(3,8)(9,12,14,13)(10,11,16,15)(17,20,23,30)(18,24,27,32)(19,29,25,22)(21,26,28,31);(1,3)(2,4)(5,8)(6,7)(9,10)(11,12)(13,15)(14,16)(17,21,23,28)(18,25,27,19)(20,31,30,26)(22,24,29,32);(1,4)(2,3)(5,6)(7,8)(9,13,14,12)(10,15,16,11)(17,22,23,29)(18,26,27,31)(19,30,25,20)(21,24,28,32)

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=h,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, 8841732);