#7763586

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

5007 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)(17,18,23,25)(19,28,20,24)(21,22,27,26)(29,32,31,30);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(17,19,23,20)(18,24,25,28)(21,29,27,31)(22,30,26,32);(2,5)(6,8)(18,25)(19,20)(22,26)(29,31);(3,7)(6,8)(9,11,13,12)(10,14,16,15)(17,20,23,19)(18,24,25,28)(21,29,27,31)(22,32,26,30);(1,3)(2,6)(4,7)(5,8)(9,11,13,12)(10,14,16,15)(17,21,23,27)(18,26,25,22)(19,29,20,31)(24,32,28,30);(1,2)(3,6)(4,5)(7,8)(9,12,13,11)(10,15,16,14)(17,22,23,26)(18,27,25,21)(19,30,20,32)(24,31,28,29)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^c=fi,e^d=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, 7670782);