#3069269

order	512
nilpotency class	2
isoclinism	isoclinic class of #3069037
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

2362 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)(9,10)(11,17)(12,20)(13,14)(15,19)(16,21)(18,23)(22,24)(25,26,30,33)(27,31,28,34)(29,37,38,32)(35,40,36,39);(6,8)(10,14)(12,21)(15,17)(18,22)(25,27,30,28)(26,31,33,34)(29,35,38,36)(32,39,37,40);(5,6)(7,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,24)(20,21)(22,23)(25,28,30,27)(26,31,33,34)(29,36,38,35)(32,39,37,40);(1,2)(3,4)(9,11,13,19)(10,15,14,17)(12,22,21,18)(16,24,20,23)(25,26)(27,34)(28,31)(29,32)(30,33)(35,40)(36,39)(37,38);(9,12)(10,16)(11,18)(13,21)(14,20)(15,23)(17,24)(19,22)(25,29)(26,32)(27,35)(28,36)(30,38)(31,39)(33,37)(34,40)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2976465);