#10086351

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
prime radical	#43015	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4517 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=ghi,d^2=i,e^2=gi,f^2=ghi,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, 9993547);