#10087714

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

5176 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)(3,7)(4,5)(6,8)(17,18)(19,22)(20,27)(21,24)(23,28)(25,29)(26,32)(30,31);(1,2,5,4)(3,6,8,7)(9,10)(11,14)(12,15)(13,16)(19,25)(22,29)(26,31)(30,32);(1,3)(2,6)(4,7)(5,8)(9,11)(10,13)(12,16)(14,15)(17,18,21,24)(19,22,25,29)(20,23,28,27)(26,30,31,32);(1,3,5,8)(2,6,4,7)(9,10)(11,13)(12,15)(14,16)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32);(1,3)(2,7)(4,6)(5,8)(9,11)(10,14)(12,16)(13,15)(17,19,21,25)(18,22,24,29)(20,26,28,31)(23,30,27,32);(1,4,5,2)(3,7,8,6)(9,10)(11,13)(12,15)(14,16)(17,20)(18,23)(19,26)(21,28)(22,30)(24,27)(25,31)(29,32)

Polycyclic presentation:

2^9|b^2=g,c^2=gh,d^2=g,e^2=gh,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, 9994910);