#8472908

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

4688 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)(9,10,15,20)(11,17,12,16)(13,21,23,18)(14,19,24,22)(28,36)(32,37)(34,40)(38,39);(2,6)(3,8)(9,11,15,12)(10,16,20,17)(13,22,23,19)(14,21,24,18)(25,26)(27,31)(28,37)(29,33)(30,35)(32,36)(34,38)(39,40);(1,3)(2,4)(5,8)(6,7)(25,27)(26,30)(28,34)(29,35)(31,33)(32,38)(36,40)(37,39);(1,3)(2,4)(5,8)(6,7)(9,12,15,11)(10,17,20,16)(13,22,23,19)(14,21,24,18)(25,27)(26,31)(28,34)(29,35)(30,33)(32,39)(36,40)(37,38);(1,4)(2,3)(5,7)(6,8)(9,13,15,23)(10,18,20,21)(11,19,12,22)(14,16,24,17)(25,28)(26,32)(27,34)(29,36)(30,38)(31,39)(33,37)(35,40);(9,14)(10,19)(11,21)(12,18)(13,16)(15,24)(17,23)(20,22)(25,26)(27,30)(28,32)(29,33)(31,35)(34,38)(36,37)(39,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=gh,e^2=gh,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, 8380104);