#7664973

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

4681 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=i,d^2=gh,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, 7572169);