#4414406

order	512
nilpotency class	2
isoclinism	isoclinic class of #4414082
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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3442 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)(25,26,31,28)(27,34,29,32)(30,33,38,36)(35,40,37,39);(9,10)(11,14)(12,15)(13,16)(17,19)(18,22)(20,21)(23,24)(25,27,31,29)(26,32,28,34)(30,35,38,37)(33,39,36,40);(5,6)(7,8)(9,10)(11,14)(12,20)(13,16)(15,21)(17,19)(18,23)(22,24)(25,28,31,26)(27,32,29,34)(30,33,38,36)(35,40,37,39);(1,2)(3,4)(5,6)(7,8)(9,11,13,19)(10,14,16,17)(12,18,21,24)(15,22,20,23)(25,29,31,27)(26,32,28,34)(30,37,38,35)(33,39,36,40);(9,12)(10,15)(11,18)(13,21)(14,22)(16,20)(17,23)(19,24)(25,30)(26,33)(27,35)(28,36)(29,37)(31,38)(32,39)(34,40)

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=h,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^c=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, 4321602);