#10094978

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

5156 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)(3,4)(5,8)(6,7)(17,18)(19,30)(20,23)(21,31)(22,27)(24,29)(25,28)(26,32);(1,3,5,7)(2,4,8,6)(9,10,13,16)(11,12,15,14)(17,19)(18,22)(20,31)(21,25)(23,32)(24,27)(26,28)(29,30);(1,3)(2,6)(4,8)(5,7)(9,11)(10,14)(12,16)(13,15)(17,20,24,28)(18,25,29,23)(19,31,27,26)(21,30,32,22);(1,3,5,7)(2,4,8,6)(9,12)(10,15)(11,16)(13,14)(17,21,24,32)(18,26,29,31)(19,25,27,23)(20,22,28,30);(1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,22)(18,27)(19,29)(20,21)(23,31)(24,30)(25,26)(28,32);(1,4)(2,7)(3,8)(5,6)(9,12)(10,15)(11,16)(13,14)(17,23,24,25)(18,28,29,20)(19,32,27,21)(22,26,30,31)

Polycyclic presentation:

2^9|b^2=i,c^2=g,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10002174);