#8088202

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

4726 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gh,d^2=i,f^2=g,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, 7995398);