#7633205

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
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4916 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7540401);