#8159294

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

4734 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)(3,7)(4,8)(5,6)(9,10,14,12)(11,16,13,15)(17,18,23,24)(19,20,27,32)(21,25,22,26)(28,30,29,31);(1,3)(2,4)(5,8)(6,7)(9,11,14,13)(10,15,12,16)(18,24)(19,27)(25,26)(28,29);(1,3)(2,4)(5,8)(6,7)(9,10,14,12)(11,15,13,16)(17,19)(18,20)(21,28)(22,29)(23,27)(24,32)(25,30)(26,31);(2,6)(3,8)(10,12)(15,16)(17,20)(18,19)(21,30)(22,31)(23,32)(24,27)(25,28)(26,29);(1,3)(2,4)(5,8)(6,7)(9,12,14,10)(11,16,13,15)(17,21,23,22)(18,25,24,26)(19,28,27,29)(20,30,32,31);(1,4)(2,3)(5,7)(6,8)(9,13,14,11)(10,16,12,15)(17,22,23,21)(18,26,24,25)(19,29,27,28)(20,31,32,30)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8066490);