#7643073

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

4906 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,d^2=g,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^c=ehi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7550269);