#8720605

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

4990 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=gi,d^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8627801);