#10095076

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:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=g,d^2=gh,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, 10002272);