#8525397

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

4750 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)(4,8)(5,6)(9,10,13,15)(11,17,19,14)(12,16,21,23)(18,24,20,22)(25,26)(27,34)(28,32)(29,31)(30,33)(35,40)(36,39)(37,38);(2,6)(3,8)(9,11,13,19)(10,14,15,17)(12,18,21,20)(16,22,23,24)(27,28)(32,34)(35,36)(39,40);(1,3)(2,4)(5,8)(6,7)(10,15)(14,17)(16,23)(22,24)(25,27,31,28)(26,32,29,34)(30,35,38,36)(33,39,37,40);(1,3)(2,4)(5,8)(6,7)(9,11)(10,14)(12,20)(13,19)(15,17)(16,24)(18,21)(22,23)(25,28,31,27)(26,32,29,34)(30,35,38,36)(33,40,37,39);(1,4)(2,3)(5,7)(6,8)(9,10)(11,14)(12,16)(13,15)(17,19)(18,22)(20,24)(21,23)(25,29,31,26)(27,32,28,34)(30,37,38,33)(35,39,36,40);(9,12)(10,16)(11,18)(13,21)(14,22)(15,23)(17,24)(19,20)(25,30,31,38)(26,33,29,37)(27,35,28,36)(32,39,34,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=gh,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=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, 8432593);