#7654946

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

5050 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=hi,e^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, 7562142);