#7977480

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

4791 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=hi,f^2=gh,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, 7884676);