#10144541

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

4755 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)(3,5)(4,6)(7,8)(17,18,23,21)(19,27,30,24)(20,31,29,25)(22,26,28,32);(1,3,7,6)(2,4,8,5)(9,10)(11,12)(13,15)(14,16)(17,19)(18,24)(20,28)(21,27)(22,29)(23,30)(25,32)(26,31);(1,4,7,5)(2,3,8,6)(9,11)(10,14)(12,15)(13,16);(2,8)(4,5)(9,11)(10,12)(13,16)(14,15)(17,20,23,29)(18,25,21,31)(19,28,30,22)(24,32,27,26);(1,5,7,4)(2,3,8,6)(9,12,13,14)(10,11,15,16)(17,21,23,18)(19,27,30,24)(20,25,29,31)(22,26,28,32);(1,6,7,3)(2,5,8,4)(9,11)(10,12)(13,16)(14,15)(17,22,23,28)(18,26,21,32)(19,29,30,20)(24,31,27,25)

Polycyclic presentation:

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

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, 10051737);