#3069168

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	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2508 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)(10,14)(11,17)(12,19)(22,24)(25,26,30,34)(27,32,28,31)(29,33,38,37)(35,40,36,39);(6,8)(11,17)(15,20)(18,23)(22,24)(25,27,30,28)(26,31,34,32)(29,36,38,35)(33,40,37,39);(5,6)(7,8)(9,10,13,14)(11,15,17,20)(12,16,19,21)(18,22,23,24)(25,28,30,27)(26,32,34,31)(29,36,38,35)(33,40,37,39);(1,2)(3,4)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,26)(27,32)(28,31)(29,37)(30,34)(33,38)(35,39)(36,40);(9,12)(10,16)(11,18)(13,19)(14,21)(15,22)(17,23)(20,24)(25,29)(26,33)(27,35)(28,36)(30,38)(31,39)(32,40)(34,37)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=g,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, 2976364);