#7665656

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4478 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=gh,e^2=hi,f^2=hi,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, 7572852);