#10158420

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
prime radical	#59407	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4432 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)(6,7)(9,10,14,15)(11,13,17,21)(12,16,19,22)(18,20,23,24)(25,26,31,29)(27,35,37,32)(28,38,36,33)(30,34,40,39);(1,2,5,4)(3,7,8,6)(12,19)(16,22)(18,23)(20,24)(25,27,31,37)(26,32,29,35)(28,30,36,40)(33,34,38,39);(1,3,5,8)(2,6,4,7)(10,15)(11,17)(16,22)(18,23);(1,4,5,2)(3,7,8,6)(9,11)(10,13)(12,18)(14,17)(15,21)(16,20)(19,23)(22,24)(25,28,31,36)(26,33,29,38)(27,30,37,40)(32,34,35,39);(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,29,31,26)(27,35,37,32)(28,33,36,38)(30,34,40,39);(9,13)(10,11)(12,20)(14,21)(15,17)(16,18)(19,24)(22,23)(25,30)(26,34)(27,36)(28,37)(29,39)(31,40)(32,38)(33,35)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=i,d^2=g,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10065616);