#8475171

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4486 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gh,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8382367);