#10075918

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

5201 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)(3,5)(4,7)(6,8)(9,10)(11,12)(13,14)(15,16)(17,18,22,25)(19,20,23,26)(21,24,30,32)(27,28,31,29);(1,3)(2,4)(5,8)(6,7)(9,11,14,15)(10,12,13,16)(17,19)(18,20)(21,27)(22,23)(24,28)(25,26)(29,32)(30,31);(1,4,6,5)(2,7,8,3)(9,12,14,16)(10,15,13,11)(19,23)(20,26)(27,31)(28,29);(1,2)(3,5)(4,7)(6,8)(9,10,14,13)(11,16,15,12)(17,18)(19,26)(20,23)(21,24)(22,25)(27,29)(28,31)(30,32);(1,2)(3,4)(5,7)(6,8)(9,12)(10,11)(13,15)(14,16)(17,20,22,26)(18,23,25,19)(21,29,30,28)(24,27,32,31);(1,5,6,4)(2,3,8,7)(9,13,14,10)(11,16,15,12)(17,21,22,30)(18,24,25,32)(19,27,23,31)(20,28,26,29)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gh,d^2=h,e^2=hi,f^2=gi,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, 9983114);