#7643313

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
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4612 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,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, 7550509);