#10366169

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

5202 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)(18,22)(23,26)(24,28)(30,32);(1,2,5,7)(3,4,8,6)(17,18,21,22)(19,23,25,26)(20,28,29,24)(27,32,31,30);(1,3,5,8)(2,6,7,4)(9,10)(11,14)(12,13)(15,16)(19,25)(23,26)(27,31)(30,32);(1,3,5,8)(2,6,7,4)(9,11,12,15)(10,14,13,16)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31);(1,3)(2,4)(5,8)(6,7)(9,11)(10,14)(12,15)(13,16)(17,19,21,25)(18,23,22,26)(20,27,29,31)(24,30,28,32);(1,4)(2,3)(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)

Polycyclic presentation:

2^9|b^2=gi,c^2=i,d^2=hi,e^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, 10273365);