#3080169

order	512
nilpotency class	2
isoclinism	isoclinic class of #3069037
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

1610 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 44 points:

(2,4)(13,14)(15,21)(16,24)(17,18)(19,23)(20,25)(22,27)(26,28)(29,30,34,32)(31,37,39,35)(33,36,42,40)(38,44,41,43);(6,9)(10,12)(14,18)(16,25)(19,21)(22,26)(29,31,34,39)(30,35,32,37)(33,41,42,38)(36,44,40,43);(5,6,8,9)(7,10,11,12)(13,14)(15,21)(16,20)(17,18)(19,23)(22,28)(24,25)(26,27)(29,31)(30,35)(32,37)(33,38)(34,39)(36,43)(40,44)(41,42);(1,2)(3,4)(13,15,17,23)(14,19,18,21)(16,26,25,22)(20,28,24,27)(29,32,34,30)(31,35,39,37)(33,36,42,40)(38,44,41,43);(5,7,8,11)(6,10,9,12)(13,16)(14,20)(15,22)(17,25)(18,24)(19,27)(21,28)(23,26)(29,33,34,42)(30,36,32,40)(31,38,39,41)(35,43,37,44)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=fh,d^2=f,e^2=fgh,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, 2987365);