#8509407

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

5188 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)(11,15)(12,16)(13,14)(17,18)(19,23)(20,27)(21,24)(22,25)(26,31)(28,32)(29,30);(1,2)(3,6)(4,5)(7,8)(10,14)(11,16)(17,19,22,27)(18,23,25,20)(21,26,30,32)(24,31,29,28);(2,5)(6,8)(9,11)(10,12)(13,16)(14,15)(17,20,22,23)(18,19,25,27)(21,28,30,31)(24,26,29,32);(3,7)(6,8)(9,11)(10,12)(13,16)(14,15)(17,18)(19,20)(21,29)(22,25)(23,27)(24,30)(26,31)(28,32);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,15)(14,16)(17,20)(18,19)(21,28)(22,23)(24,26)(25,27)(29,32)(30,31);(1,3)(2,6)(4,7)(5,8)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,29)(27,32)

Polycyclic presentation:

2^9|b^2=gh,c^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=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, 8416603);