#8664713

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	#40780	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4566 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=ghi,d^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8571909);