#7643492

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

5446 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=gh,d^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, 7550688);