#8618251

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

4979 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)(3,4)(5,8)(6,7)(10,13)(11,15)(17,18)(19,25)(20,29)(21,31)(22,24)(23,28)(26,27)(30,32);(1,3)(2,4)(5,6)(7,8)(9,10,12,13)(11,14,15,16)(17,19)(18,23)(20,30)(21,32)(22,28)(24,25)(26,31)(27,29);(1,4)(2,5)(3,7)(6,8)(9,10)(11,14)(12,13)(15,16)(18,24)(21,29)(23,25)(27,32);(1,3,6,5)(2,4,7,8)(9,10)(11,14)(12,13)(15,16)(17,20)(18,21)(19,26)(22,31)(23,27)(24,29)(25,32)(28,30);(1,5,6,3)(2,4,7,8)(9,11,12,15)(10,14,13,16)(17,21)(18,20)(19,27)(22,29)(23,26)(24,31)(25,30)(28,32);(1,4)(2,3)(5,7)(6,8)(17,19)(18,25)(20,30)(21,27)(22,28)(23,24)(26,31)(29,32)

Polycyclic presentation:

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

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, 8525447);