#7653641

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=h,e^2=gi,f^2=gi,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, 7560837);