#7733972

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

4869 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7641168);