#7973876

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

4982 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=i,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, 7881072);