#8091721

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

5094 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)(3,6)(4,5)(7,8)(9,10,13,16)(11,14,12,15)(17,18,22,25)(19,26,27,23)(20,21,24,31)(28,29,30,32);(9,10)(11,14)(12,15)(13,16)(17,19)(18,23)(20,30)(21,29)(22,27)(24,28)(25,26)(31,32);(2,5)(6,8)(19,27)(23,26)(28,30)(29,32);(3,7)(6,8)(17,20)(18,21)(19,28)(22,24)(23,32)(25,31)(26,29)(27,30);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,14,16,15)(17,19)(18,23)(20,30)(21,29)(22,27)(24,28)(25,26)(31,32);(1,2)(3,6)(4,5)(7,8)(9,12,13,11)(10,15,16,14)(17,21)(18,24)(19,29)(20,25)(22,31)(23,28)(26,30)(27,32)

Polycyclic presentation:

2^9|a^2=g,e^2=ghi,f^2=gi,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, 7998917);