#10521321

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

4865 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)(3,8)(4,6)(5,7)(9,10)(11,15)(12,14)(13,16)(17,18)(19,24)(20,30)(21,25)(22,31)(23,27)(26,28)(29,32);(1,3)(2,4)(5,6)(7,8)(9,10,12,14)(11,13,16,15)(17,19,23,29)(18,24,27,32)(20,28,21,22)(25,31,30,26);(9,10)(11,13)(12,14)(15,16)(17,20,23,21)(18,25,27,30)(19,28,29,22)(24,31,32,26);(1,2)(3,8)(4,6)(5,7)(9,11,12,16)(10,13,14,15)(17,21,23,20)(18,25,27,30)(19,22,29,28)(24,31,32,26);(1,4,5,8)(2,6,7,3)(17,18)(19,24)(20,25)(21,30)(22,26)(23,27)(28,31)(29,32);(1,2)(3,4)(5,7)(6,8)(17,22)(18,26)(19,20)(21,29)(23,28)(24,25)(27,31)(30,32)

Polycyclic presentation:

2^9|b^2=hi,c^2=h,d^2=hi,e^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10428517);