#7722062

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

5730 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)(11,16)(12,14)(13,15)(17,18)(19,28)(20,26)(21,31)(22,27)(23,25)(24,29)(30,32);(1,2)(3,7)(4,5)(6,8)(9,11,13,12)(10,14,15,16)(17,19)(18,24)(20,22)(21,26)(23,29)(25,28)(27,32)(30,31);(2,5)(6,7)(11,12)(14,16)(18,25)(19,29)(20,30)(21,27);(17,20,23,30)(18,26,25,32)(19,22,29,31)(21,28,27,24);(1,2)(3,6)(4,5)(7,8)(9,12,13,11)(10,14,15,16)(17,21,23,27)(18,22,25,31)(19,26,29,32)(20,28,30,24);(1,3)(2,6)(4,8)(5,7)(9,10)(11,16)(12,14)(13,15)(17,22)(18,27)(19,30)(20,29)(21,25)(23,31)(24,26)(28,32)

Polycyclic presentation:

2^9|b^2=gh,d^2=h,e^2=g,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, 7629258);