#10408104

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

4400 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,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, 10315300);