#10521364

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

4558 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)(9,10,15,19)(11,18,22,16)(12,14,13,17)(20,21,23,24)(25,26,30,33)(27,34,29,31)(28,32,37,40)(35,39,36,38);(9,11)(10,16)(12,23)(13,20)(14,21)(15,22)(17,24)(18,19)(25,27,30,29)(26,31,33,34)(28,36,37,35)(32,39,40,38);(1,2)(3,6)(4,5)(7,8)(9,12,15,13)(10,17,19,14)(11,20,22,23)(16,21,18,24)(25,27,30,29)(26,31,33,34)(28,35,37,36)(32,38,40,39);(3,7)(6,8)(9,13,15,12)(10,14,19,17)(11,20,22,23)(16,21,18,24)(25,28)(26,32)(27,35)(29,36)(30,37)(31,38)(33,40)(34,39);(1,3,4,7)(2,6,5,8)(9,11)(10,18)(12,20)(13,23)(14,21)(15,22)(16,19)(17,24)(25,29,30,27)(26,31,33,34)(28,35,37,36)(32,39,40,38);(1,2)(3,6)(4,5)(7,8)(9,14)(10,12)(11,21)(13,19)(15,17)(16,23)(18,20)(22,24)(25,26)(27,31)(28,32)(29,34)(30,33)(35,38)(36,39)(37,40)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=g,d^2=i,e^2=hi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10428560);