#7643014

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	#39359	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5124 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=h,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^c=ehi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7550210);