#7966871

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

5432 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,d^2=g,e^2=gh,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, 7874067);