#8073171

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

4869 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:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=h,e^2=g,f^2=g,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, 7980367);