#7692423

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

4907 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)(11,14)(12,16)(13,15)(17,18)(19,28)(20,31)(21,26)(22,25)(23,27)(24,30)(29,32);(1,2)(3,5)(4,6)(7,8)(9,11,13,12)(10,14,15,16)(17,19)(18,24)(20,21)(22,29)(23,30)(25,26)(27,28)(31,32);(9,12,13,11)(10,14,15,16)(17,20,23,22)(18,25,27,31)(19,29,30,21)(24,32,28,26);(9,10)(11,14)(12,16)(13,15)(17,21,23,29)(18,26,27,32)(19,20,30,22)(24,25,28,31);(1,3,4,7)(2,5,6,8)(17,22,23,20)(18,25,27,31)(19,29,30,21)(24,26,28,32);(17,18)(19,24)(20,25)(21,26)(22,31)(23,27)(28,30)(29,32)

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=h,d^2=i,e^2=g,b^a=bg,c^a=ch,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7599619);