#3071442

order	512
nilpotency class	2
isoclinism	isoclinic class of #3069037
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

2118 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=gh,d^2=f,e^2=fg,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, 2978638);