#4419436

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

3414 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)(11,17)(12,19)(14,20)(15,22)(25,26,31,28)(27,34,32,29)(30,33,39,36)(35,38,40,37);(9,10,13,16)(11,14,17,20)(12,15,19,22)(18,21,23,24)(25,27)(26,29)(28,34)(30,35)(31,32)(33,37)(36,38)(39,40);(5,6)(7,8)(9,10)(11,14)(12,15)(13,16)(17,20)(18,21)(19,22)(23,24)(25,28,31,26)(27,34,32,29)(30,36,39,33)(35,38,40,37);(1,2)(3,4)(5,6)(7,8)(9,11,13,17)(10,14,16,20)(12,18,19,23)(15,21,22,24)(25,29,31,34)(26,32,28,27)(30,38,39,37)(33,35,36,40);(9,12)(10,15)(11,18)(13,19)(14,21)(16,22)(17,23)(20,24)(25,30,31,39)(26,33,28,36)(27,35,32,40)(29,37,34,38)

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=f,d^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, 4326632);