#10329291

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

4742 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:

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

Polycyclic presentation:

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

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, 10236487);