#7659045

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

4720 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=i,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^b=dhi,f^e=fhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7566241);