#7649174

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

5060 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,e^d=ehi

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, 7556370);