#7987949

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

4918 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)(9,10,13,15)(11,12,16,14)(17,18,23,25)(19,27,28,24)(20,30,29,26)(21,31,32,22);(1,2)(3,7)(4,8)(5,6)(17,19)(18,24)(20,22)(21,26)(23,28)(25,27)(29,31)(30,32);(1,3,5,4)(2,7,6,8)(9,11,13,16)(10,14,15,12)(18,25)(19,28)(20,29)(21,32);(1,3,5,4)(2,7,6,8)(9,11)(10,12)(13,16)(14,15)(17,20,23,29)(18,26,25,30)(19,22,28,31)(21,27,32,24);(9,12)(10,11)(13,14)(15,16)(17,21,23,32)(18,22,25,31)(19,26,28,30)(20,27,29,24);(1,4,5,3)(2,8,6,7)(17,22)(18,21)(19,29)(20,28)(23,31)(24,30)(25,32)(26,27)

Polycyclic presentation:

2^9|a^2=h,c^2=gh,d^2=g,e^2=i,f^2=gi,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, 7895145);