#8003821

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

4572 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)(3,6)(4,5)(7,8)(9,10,13,16)(11,14,12,15)(17,18,22,25)(19,23,29,26)(20,21,24,30)(27,28,31,32);(1,3,5,8)(2,6,4,7)(9,10)(11,14)(12,15)(13,16)(17,19)(18,23)(20,27)(21,28)(22,29)(24,31)(25,26)(30,32);(1,3)(2,7)(4,6)(5,8)(9,11,13,12)(10,14,16,15)(17,18,22,25)(19,26,29,23)(20,21,24,30)(27,32,31,28);(1,3,5,8)(2,6,4,7)(9,11,13,12)(10,14,16,15)(17,20)(18,21)(19,27)(22,24)(23,28)(25,30)(26,32)(29,31);(1,3)(2,6)(4,7)(5,8)(9,11,13,12)(10,14,16,15)(17,21)(18,24)(19,28)(20,25)(22,30)(23,31)(26,27)(29,32);(1,4,5,2)(3,7,8,6)(9,12,13,11)(10,15,16,14)(17,19)(18,23)(20,27)(21,28)(22,29)(24,31)(25,26)(30,32)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=ghi,d^2=gh,e^2=g,f^2=gh,b^a=bg,c^a=ch,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fi,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7911017);