#8757502

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
prime radical	#45451	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4260 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=gi,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8664698);