#7653853

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

5810 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:

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

Polycyclic presentation:

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

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, 7561049);