#8702621

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

5168 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=h,d^2=h,e^2=g,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, 8609817);