#7968548

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

5231 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)(9,10,13,16)(11,12,14,15)(17,18,22,24)(19,27,28,23)(20,31,30,25)(21,26,29,32);(2,5)(7,8)(9,11)(10,12)(13,14)(15,16)(17,19,22,28)(18,23,24,27)(20,29,30,21)(25,32,31,26);(1,2,6,5)(3,7,4,8)(9,12)(10,14)(11,16)(13,15)(18,24)(19,28)(20,30)(26,32);(1,3,6,4)(2,7,5,8)(9,11)(10,15)(12,16)(13,14)(17,18)(19,23)(20,25)(21,26)(22,24)(27,28)(29,32)(30,31);(1,4,6,3)(2,7,5,8)(9,11)(10,12)(13,14)(15,16)(17,20)(18,25)(19,29)(21,28)(22,30)(23,32)(24,31)(26,27);(1,5,6,2)(3,8,4,7)(9,12)(10,14)(11,16)(13,15)(17,21,22,29)(18,26,24,32)(19,30,28,20)(23,31,27,25)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gh,d^2=gh,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, 7875744);