#7888622

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

4733 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)(3,4)(5,6)(7,8)(9,10,12,14)(11,13,15,16)(17,18)(19,29)(20,25)(21,26)(22,27)(23,28)(24,32)(30,31);(1,3,7,5)(2,4,8,6)(9,10)(11,13)(12,14)(15,16)(17,19)(18,24)(20,30)(21,31)(22,26)(23,32)(25,27)(28,29);(1,4,7,6)(2,5,8,3)(17,18)(19,29)(20,25)(21,26)(22,27)(23,28)(24,32)(30,31);(1,3,7,5)(2,4,8,6)(9,11,12,15)(10,13,14,16)(17,20,23,26)(18,25,28,21)(19,30,32,22)(24,27,29,31);(1,5,7,3)(2,4,8,6)(17,21,23,25)(18,26,28,20)(19,31,32,27)(22,29,30,24);(1,6,7,4)(2,5,8,3)(17,22,23,30)(18,27,28,31)(19,26,32,20)(21,24,25,29)

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7795818);