#8127498

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

4412 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=i,d^2=h,e^2=gh,f^2=g,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^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, 8034694);