#8701522

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	#2676	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4272 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8608718);