#7633297

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	C₂×#532	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4996 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=hi,d^2=g,f^2=i,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, 7540493);