#7653482

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	#35191	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4778 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=h,e^2=ghi,f^2=hi,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, 7560678);