#8733505

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

4885 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)(3,7)(4,5)(6,8)(9,10,12,13)(11,15,16,14)(17,18)(19,24)(20,31)(21,23)(22,26)(25,29)(27,32)(28,30);(1,3)(2,6)(4,7)(5,8)(10,13)(14,15)(17,19,23,30)(18,24,21,28)(20,22,29,27)(25,32,31,26);(1,3)(2,6)(4,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,20,23,29)(18,25,21,31)(19,27,30,22)(24,26,28,32);(3,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,20)(18,25)(19,27)(21,31)(22,30)(23,29)(24,26)(28,32);(1,4,5,2)(3,7,8,6)(9,11,12,16)(10,14,13,15)(17,21,23,18)(19,28,30,24)(20,31,29,25)(22,26,27,32);(1,3)(2,7)(4,6)(5,8)(17,22,23,27)(18,26,21,32)(19,29,30,20)(24,25,28,31)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8640701);