#7714526

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

5044 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)(3,7)(4,6)(5,8)(9,10,13,12)(11,15,16,14)(17,18,22,24)(19,21,26,23)(20,28,29,25)(27,31,32,30);(9,11,13,16)(10,14,12,15)(17,19,22,26)(18,23,24,21)(20,27,29,32)(25,31,28,30);(1,2,4,6)(3,5,8,7)(9,12,13,10)(11,15,16,14)(18,24)(19,26)(20,29)(30,31);(1,2)(3,5)(4,6)(7,8)(17,19)(18,21)(20,27)(22,26)(23,24)(25,30)(28,31)(29,32);(1,3)(2,5)(4,8)(6,7)(17,20,22,29)(18,25,24,28)(19,27,26,32)(21,30,23,31);(17,21)(18,19)(20,30)(22,23)(24,26)(25,27)(28,32)(29,31)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,e^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7621722);