#7653748

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	#39725	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4759 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^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, 7560944);