#7655312

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

4882 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=hi,d^2=g,e^2=gh,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, 7562508);