#8472956

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

4492 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=gh,e^2=h,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, 8380152);