#7664908

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

4784 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)(10,14)(15,16)(17,18)(19,26)(20,28)(21,24)(22,23)(25,30)(27,31)(29,32);(1,3,5,6)(2,4,7,8)(9,10,13,14)(11,15,12,16)(17,19,23,21)(18,24,22,26)(20,27,30,29)(25,31,28,32);(1,4)(2,6)(3,7)(5,8)(9,11,13,12)(10,15,14,16)(19,21)(24,26)(27,29)(31,32);(17,20)(18,25)(19,27)(21,29)(22,28)(23,30)(24,31)(26,32);(1,2)(3,4)(5,7)(6,8)(9,11,13,12)(10,15,14,16)(17,21,23,19)(18,24,22,26)(20,29,30,27)(25,31,28,32);(1,4)(2,6)(3,7)(5,8)(9,12,13,11)(10,16,14,15)(17,22,23,18)(19,26,21,24)(20,28,30,25)(27,32,29,31)

Polycyclic presentation:

2^9|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, 7572104);