#9726605

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

4912 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:

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

Polycyclic presentation:

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

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, 9633801);