#7643351

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	#37276	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4725 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:

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

Polycyclic presentation:

2^9|b^2=h,d^2=gh,e^2=g,f^2=hi,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, 7550547);