#8073149

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

4929 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)(4,7)(10,14)(15,16)(17,18)(19,20)(21,25)(22,24)(23,26)(27,28)(29,32)(30,31);(1,2,5,6)(3,7,8,4)(9,10)(11,15)(12,16)(13,14)(17,19)(18,21)(20,22)(23,27)(24,25)(26,29)(28,30)(31,32);(2,6)(4,7)(17,20,24,21)(18,25,22,19)(23,29,31,28)(26,27,30,32);(1,3,5,8)(2,7,6,4)(9,11,13,12)(10,15,14,16)(17,21,24,20)(18,25,22,19)(23,28,31,29)(26,27,30,32);(1,4,5,7)(2,3,6,8)(9,12,13,11)(10,16,14,15)(17,22,24,18)(19,20,25,21)(23,30,31,26)(27,28,32,29);(1,3)(2,7)(4,6)(5,8)(17,23)(18,26)(19,27)(20,28)(21,29)(22,30)(24,31)(25,32)

Polycyclic presentation:

2^9|b^2=hi,c^2=h,d^2=g,e^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, 7980345);