#8810069

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	#37093	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4185 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gh,d^2=g,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,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, 8717265);