#8108789

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

4964 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:

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

Polycyclic presentation:

2^9|c^2=ghi,d^2=gh,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi

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, 8015985);