#7656789

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

4599 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)(3,8)(4,6)(5,7)(10,14)(15,16)(17,18)(19,21)(20,28)(22,23)(24,26)(25,30)(27,31)(29,32);(9,10)(11,15)(12,16)(13,14)(18,23)(19,26)(25,28)(27,32);(1,3,5,6)(2,4,7,8)(17,19)(18,24)(20,27)(21,23)(22,26)(25,31)(28,29)(30,32);(9,11,13,12)(10,15,14,16)(17,20,22,30)(18,25,23,28)(19,27,26,32)(21,29,24,31);(1,2)(3,4)(5,7)(6,8)(9,12,13,11)(10,16,14,15);(1,4)(2,6)(3,7)(5,8)(17,21,22,24)(18,19,23,26)(20,29,30,31)(25,27,28,32)

Polycyclic presentation:

2^9|c^2=hi,d^2=g,e^2=gi,f^2=i,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, 7563985);