#10023314

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

5173 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)(3,7)(4,6)(5,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,29)(28,31)(30,32);(1,2,4,6)(3,5,8,7)(9,11,14,15)(10,12,13,16)(18,23)(19,26)(25,31)(27,32);(1,2,4,6)(3,5,8,7)(9,12,14,16)(10,15,13,11)(20,28)(25,31)(27,32)(29,30);(1,2)(3,5)(4,6)(7,8)(9,10,14,13)(11,16,15,12)(17,19,22,26)(18,24,23,21)(20,27,28,32)(25,29,31,30);(1,2)(3,7)(4,6)(5,8)(9,12)(10,11)(13,15)(14,16)(17,20)(18,25)(19,27)(21,30)(22,28)(23,31)(24,29)(26,32);(1,3)(2,5)(4,8)(6,7)(9,13,14,10)(11,16,15,12)(17,21,22,24)(18,26,23,19)(20,29,28,30)(25,27,31,32)

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,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, 9930510);