#10040630

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

4513 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,d^2=hi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,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, 9947826);