#10156265

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

4878 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=ghi,d^2=h,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, 10063461);