#8075365

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

4908 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)(3,6)(4,5)(7,8)(9,10,13,12)(11,15,16,14)(17,18,23,24)(19,25,21,20)(22,26,30,32)(27,31,29,28);(9,11,13,16)(10,14,12,15)(17,19,23,21)(18,20,24,25)(22,27,30,29)(26,28,32,31);(2,5)(6,8)(9,10,13,12)(11,14,16,15)(18,24)(19,21)(22,30)(28,31);(3,7)(6,8)(9,12,13,10)(11,15,16,14)(17,20,23,25)(18,19,24,21)(22,31,30,28)(26,29,32,27);(1,3,4,7)(2,6,5,8)(17,21,23,19)(18,25,24,20)(22,29,30,27)(26,31,32,28);(1,2)(3,6)(4,5)(7,8)(17,22)(18,26)(19,27)(20,28)(21,29)(23,30)(24,32)(25,31)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gi,d^2=g,e^2=h,b^a=bg,c^a=ch,e^a=ei,c^b=ci,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, 7982561);