#7664909

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×#537	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4830 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)(3,4)(5,7)(6,8)(9,10,13,12)(11,15,16,14)(17,18,23,26)(19,27,21,24)(20,22,31,25)(28,32,30,29);(1,3,5,6)(2,4,7,8)(9,11,13,16)(10,14,12,15)(17,19,23,21)(18,24,26,27)(20,28,31,30)(22,29,25,32);(1,4)(2,6)(3,7)(5,8)(9,10,13,12)(11,14,16,15)(19,21)(24,27)(28,30)(29,32);(17,20)(18,25)(19,28)(21,30)(22,26)(23,31)(24,32)(27,29);(1,2)(3,4)(5,7)(6,8)(9,10,13,12)(11,14,16,15)(17,21,23,19)(18,24,26,27)(20,30,31,28)(22,29,25,32);(1,4)(2,6)(3,7)(5,8)(9,12,13,10)(11,15,16,14)(17,22,23,25)(18,20,26,31)(19,29,21,32)(24,28,27,30)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=gi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fhi,c^b=ci,e^c=ehi,f^e=fhi

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7572105);