#7664845

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	#39165	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4635 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=g,e^2=gh,f^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, 7572041);