#8497807

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

4383 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)(3,7)(4,5)(6,8)(9,10)(11,20)(12,18)(13,17)(14,16)(15,19)(21,23)(22,24)(28,36)(32,37)(35,40)(38,39);(1,3)(2,6)(4,7)(5,8)(9,11,15,14)(10,16,19,20)(12,21,13,22)(17,23,18,24)(25,26)(27,31)(28,37)(29,33)(30,34)(32,36)(35,38)(39,40);(9,12,15,13)(10,17,19,18)(11,21,14,22)(16,23,20,24)(25,27)(26,29)(28,35)(30,33)(31,34)(32,38)(36,40)(37,39);(1,2,5,4)(3,6,8,7)(9,13,15,12)(10,18,19,17)(11,21,14,22)(16,23,20,24)(25,27)(26,31)(28,35)(29,34)(30,33)(32,39)(36,40)(37,38);(1,4,5,2)(3,6,8,7)(9,10)(11,20)(12,17)(13,18)(14,16)(15,19)(21,24)(22,23)(25,28)(26,32)(27,35)(29,38)(30,36)(31,39)(33,40)(34,37);(1,3)(2,7)(4,6)(5,8)(9,14,15,11)(10,16,19,20)(12,22,13,21)(17,23,18,24)(25,29,30,31)(26,33,34,27)(28,38,36,39)(32,40,37,35)

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=g,e^2=gh,f^2=hi,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, 8405003);