#8097742

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

5213 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=i,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8004938);