#7659590

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

4963 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gh,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dh,c^b=ci,d^c=dhi,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, 7566786);