#4418764

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
automorphism	Aut(G) order 8388608
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3522 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef

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, 4325960);