#7969370

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	#43888	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4734 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,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, 7876566);