#7972702

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

4888 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)(7,8)(10,12)(14,15)(17,18)(19,24)(20,30)(21,23)(22,26)(25,29)(27,28)(31,32);(1,2,5,6)(3,7,4,8)(17,19)(18,24)(20,22)(21,28)(23,27)(25,31)(26,30)(29,32);(1,3,5,4)(2,7,6,8)(9,10,13,12)(11,15,16,14)(18,21)(19,27)(20,29)(26,31);(1,3,5,4)(2,7,6,8)(9,11,13,16)(10,14,12,15)(17,20,23,29)(18,25,21,30)(19,22,27,32)(24,31,28,26);(1,3,5,4)(2,7,6,8)(9,12,13,10)(11,15,16,14)(17,21,23,18)(19,28,27,24)(20,30,29,25)(22,26,32,31);(1,4,5,3)(2,8,6,7)(17,22)(18,26)(19,29)(20,27)(21,31)(23,32)(24,25)(28,30)

Polycyclic presentation:

2^9|b^2=g,c^2=ghi,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7879898);