#8196982

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

5067 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:

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

Polycyclic presentation:

2^9|b^2=ghi,d^2=gi,e^2=h,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

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, 8104178);