#8188195

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	#38950	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4368 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=g,d^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8095391);