#7643412

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

5328 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)(3,5)(4,7)(6,8)(9,10,12,14)(11,15,16,13)(18,23)(21,31)(24,30)(25,32);(2,3)(6,7)(11,16)(13,15)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,30)(29,32);(1,3,5,2)(4,7,8,6)(9,11,12,16)(10,13,14,15)(18,23)(20,29)(24,30)(26,27);(1,4,5,8)(2,6,3,7)(9,10)(11,13)(12,14)(15,16)(17,19,22,28)(18,24,23,30)(20,27,29,26)(21,25,31,32);(17,20)(18,25)(19,27)(21,30)(22,29)(23,32)(24,31)(26,28);(17,21,22,31)(18,26,23,27)(19,25,28,32)(20,30,29,24)

Polycyclic presentation:

2^9|a^2=gi,c^2=h,d^2=g,f^2=hi,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, 7550608);