#8504751

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

4342 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)(3,5)(4,7)(6,8)(10,12)(15,18)(16,20)(22,24)(25,26)(27,35)(28,32)(29,33)(30,34)(31,38)(36,40)(37,39);(2,3)(6,7)(11,17)(15,18)(19,23)(22,24)(25,26)(27,35)(28,39)(29,40)(30,34)(31,38)(32,37)(33,36);(1,2,5,3)(4,6,8,7)(9,10,14,12)(11,15,17,18)(13,16,21,20)(19,22,23,24)(25,27)(26,31)(28,36)(29,37)(30,38)(32,33)(34,35)(39,40);(1,3,5,2)(4,7,8,6)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,28,30,37)(26,32,34,39)(27,36,38,29)(31,33,35,40);(1,4,5,8)(2,6,3,7)(9,12,14,10)(11,18,17,15)(13,20,21,16)(19,24,23,22)(25,29,30,36)(26,33,34,40)(27,37,38,28)(31,32,35,39);(9,13,14,21)(10,16,12,20)(11,19,17,23)(15,22,18,24)(25,26)(27,31)(28,39)(29,33)(30,34)(32,37)(35,38)(36,40)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8411947);