#7722755

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

4834 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:

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

Polycyclic presentation:

2^9|b^2=h,d^2=g,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei

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, 7629951);