#1714983

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

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

1536 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=fg,d^2=hi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df

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, 1622179);