#8295522

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

4274 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,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, 8202718);