#8464251

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

4558 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(1,2)(3,8)(4,6)(5,7)(10,16)(11,12)(18,20)(19,22)(25,26)(27,31)(28,38)(29,34)(30,35)(32,36)(33,37)(39,40);(1,3,5,4)(2,6,7,8)(9,10)(11,17)(12,21)(13,18)(14,19)(15,16)(20,23)(22,24)(25,27,32,30)(26,31,36,35)(28,39,37,29)(33,34,38,40);(9,11,15,12)(10,17,16,21)(13,19,23,22)(14,20,24,18)(25,28)(26,33)(27,29)(30,39)(31,40)(32,37)(34,35)(36,38);(2,7)(6,8)(9,12,15,11)(10,17,16,21)(13,22,23,19)(14,20,24,18)(25,29,32,39)(26,34,36,40)(27,37,30,28)(31,33,35,38);(1,4,5,3)(2,8,7,6)(9,13,15,23)(10,18,16,20)(11,19,12,22)(14,21,24,17)(25,30,32,27)(26,35,36,31)(28,39,37,29)(33,34,38,40);(1,2)(3,6)(4,8)(5,7)(9,14)(10,19)(11,20)(12,18)(13,21)(15,24)(16,22)(17,23)(25,31,32,35)(26,27,36,30)(28,40,37,34)(29,38,39,33)

Polycyclic presentation:

2^9|b^2=gh,c^2=hi,d^2=h,e^2=gi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,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, 8371447);