#8120060

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

4414 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=gh,d^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=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, 8027256);