#8202387

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

4778 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:

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

Polycyclic presentation:

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

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, 8109583);