#7645890

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

4571 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)(10,14)(15,16)(19,25)(20,28)(23,26)(24,29);(1,2,5,4)(3,6,8,7)(9,10,13,14)(11,15,12,16)(17,18,22,21)(19,26,25,23)(20,24,28,29)(27,32,31,30);(1,2,5,4)(3,6,8,7)(9,11,13,12)(10,15,14,16)(17,19,22,25)(18,23,21,26)(20,27,28,31)(24,30,29,32);(1,2,5,4)(3,7,8,6)(9,11,13,12)(10,15,14,16)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32);(1,3,5,8)(2,6,4,7)(9,12,13,11)(10,16,14,15);(1,4,5,2)(3,7,8,6)(17,21,22,18)(19,26,25,23)(20,29,28,24)(27,32,31,30)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7553086);