#10075931

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

5426 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)(3,6)(4,8)(5,7)(9,10)(11,12)(13,14)(15,16)(17,18,22,23)(19,24,26,21)(20,25,28,31)(27,29,32,30);(1,2,5,7)(3,8,4,6)(9,11,14,15)(10,12,13,16)(18,23)(19,26)(25,31)(27,32);(1,2,5,7)(3,8,4,6)(9,12,14,16)(10,15,13,11)(20,28)(25,31)(27,32)(29,30);(1,3,5,4)(2,6,7,8)(9,10,14,13)(11,16,15,12)(17,19)(18,24)(20,27)(21,23)(22,26)(25,29)(28,32)(30,31);(1,2)(3,6)(4,8)(5,7)(9,12)(10,11)(13,15)(14,16)(17,20,22,28)(18,25,23,31)(19,27,26,32)(21,30,24,29);(1,4,5,3)(2,6,7,8)(9,13,14,10)(11,16,15,12)(17,21,22,24)(18,19,23,26)(20,29,28,30)(25,32,31,27)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=h,d^2=h,e^2=ghi,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, 9983127);