#1148988

order	512
nilpotency class	2
isoclinism	isoclinic class of #1082705
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

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

1570 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1056184);