#8844528

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

4591 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=gh,d^2=ghi,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

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, 8751724);