#7653058

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

4781 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:

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

Polycyclic presentation:

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

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, 7560254);