#7918963

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

5086 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)(9,10,13,14)(11,15,16,12)(17,18)(19,28)(20,31)(21,32)(22,27)(23,24)(25,30)(26,29);(1,2)(3,7)(4,8)(5,6)(9,11)(10,12)(13,16)(14,15)(17,19)(18,22)(20,29)(21,30)(23,27)(24,28)(25,26)(31,32);(1,3,5,4)(2,7,6,8)(10,14)(12,15)(18,24)(19,27)(20,30)(26,32);(1,3,5,4)(2,7,6,8)(9,12)(10,11)(13,15)(14,16)(17,20,23,30)(18,25,24,31)(19,29,27,21)(22,26,28,32);(9,11)(10,12)(13,16)(14,15)(17,21,23,29)(18,26,24,32)(19,30,27,20)(22,25,28,31);(1,4,5,3)(2,8,6,7)(17,22,23,28)(18,27,24,19)(20,26,30,32)(21,31,29,25)

Polycyclic presentation:

2^9|a^2=hi,c^2=gh,d^2=ghi,e^2=i,f^2=ghi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=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, 7826159);