#8757809

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

5148 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:

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

Polycyclic presentation:

2^9|b^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8665005);