#4062924

order	512
nilpotency class	2
isoclinism	isoclinic class of #4033837
automorphism	Aut(G) order 1048576
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

1428 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=fgi,c^2=h,d^2=fh,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3970120);