#7656830

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

5633 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7564026);