#8078315

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

5114 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)(3,5)(4,6)(7,8)(10,13)(14,16)(17,18)(19,20)(21,23)(22,25)(24,26)(27,28)(29,30)(31,32);(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,21)(22,27)(23,26)(25,31)(28,29)(30,32);(1,2,4,6)(3,5,7,8)(9,11,12,15)(10,14,13,16)(18,21)(19,26)(22,30)(28,31);(3,7)(5,8)(17,20)(18,19)(21,26)(22,31)(23,24)(25,32)(27,29)(28,30);(1,3,4,7)(2,5,6,8)(17,21,23,18)(19,20,26,24)(22,29,30,25)(27,28,32,31);(1,2)(3,5)(4,6)(7,8)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)

Polycyclic presentation:

2^9|c^2=gh,e^2=h,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,e^c=ei,f^c=fh,e^d=ehi,f^d=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, 7985511);