#7969371

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
prime radical	#59060	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4731 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)(11,15)(12,13)(14,16)(17,18)(19,28)(20,30)(21,29)(22,25)(23,24)(26,31)(27,32);(1,2,5,6)(3,7,4,8)(18,24)(19,22)(26,30)(27,29);(1,3,5,4)(2,7,6,8)(10,13)(14,15)(17,19,23,22)(18,25,24,28)(20,27,31,29)(21,26,32,30);(1,3,5,4)(2,7,6,8)(9,10)(11,14)(12,13)(15,16)(17,20,23,31)(18,26,24,30)(19,29,22,27)(21,28,32,25);(1,3,5,4)(2,7,6,8)(9,11)(10,14)(12,16)(13,15)(17,21,23,32)(18,27,24,29)(19,30,22,26)(20,28,31,25);(1,4,5,3)(2,8,6,7)(17,22,23,19)(18,25,24,28)(20,27,31,29)(21,30,32,26)

Polycyclic presentation:

2^9|b^2=gi,c^2=g,d^2=g,e^2=g,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, 7876567);