#7905413

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

5089 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=ghi,e^2=gh,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7812609);