#7642739

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

4740 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=gi,e^2=gh,f^2=g,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, 7549935);