#8475202

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

4863 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)(6,8)(9,10)(11,15)(12,13)(14,17)(16,20)(18,23)(19,21)(22,24)(25,26,30,31)(27,28,33,32)(29,34,38,37)(35,36,40,39);(3,7)(6,8)(9,11,14,20)(10,15,17,16)(12,21,22,18)(13,19,24,23)(26,31)(28,32)(34,37)(36,39);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,18)(13,19)(14,20)(15,17)(21,22)(23,24)(25,27,30,33)(26,32,31,28)(29,35,38,40)(34,39,37,36);(1,3)(2,6)(4,7)(5,8)(9,11)(10,15)(12,18)(13,23)(14,20)(16,17)(19,24)(21,22)(25,26)(27,32)(28,33)(29,37)(30,31)(34,38)(35,36)(39,40);(1,2)(3,6)(4,5)(7,8)(9,12)(10,13)(11,18)(14,22)(15,23)(16,19)(17,24)(20,21)(25,28,30,32)(26,33,31,27)(29,36,38,39)(34,40,37,35);(9,13)(10,12)(11,19)(14,24)(15,21)(16,18)(17,22)(20,23)(25,29)(26,34)(27,35)(28,36)(30,38)(31,37)(32,39)(33,40)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=g,e^2=g,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, 8382398);