#8137018

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	#57199	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4438 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)(3,8,6,4)(12,20)(16,23)(18,24)(21,22)(25,26)(27,34)(28,31)(29,36)(30,33)(32,38)(35,37)(39,40);(1,3)(2,4)(5,6)(7,8)(9,10)(11,15)(12,16)(13,17)(14,19)(18,22)(20,23)(21,24)(26,28)(27,33)(32,36)(35,39);(9,11)(10,14)(12,18)(13,19)(15,17)(16,21)(20,24)(22,23)(25,27)(26,30)(28,34)(29,35)(31,33)(32,37)(36,40)(38,39);(1,4,5,8)(2,6,7,3)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,28,31,26)(27,34,33,30)(29,36,38,32)(35,40,39,37);(1,4)(2,3)(5,8)(6,7)(9,10)(11,14)(12,16)(13,17)(15,19)(18,21)(20,23)(22,24)(25,29,31,38)(26,32,28,36)(27,35,33,39)(30,37,34,40);(1,3)(2,4)(5,6)(7,8)(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)(25,30,31,34)(26,33,28,27)(29,37,38,40)(32,39,36,35)

Polycyclic presentation:

2^9|a^2=gh,d^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=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, 8044214);