#5073804

order	512
nilpotency class	2
isoclinism	isoclinic class of #5035983
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

1370 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)(3,7)(4,6)(5,8)(9,10,14,12)(11,16,13,15)(17,18,21,19)(20,23,24,22)(25,26)(27,33)(28,36)(29,30)(31,34)(32,37)(35,38)(39,40);(1,3,4,8)(2,5,6,7)(9,10,14,12)(11,15,13,16)(17,19,21,18)(20,23,24,22)(26,29)(27,34)(28,37)(38,39);(2,6)(5,7)(9,11,14,13)(10,15,12,16)(25,27)(26,31)(28,35)(29,33)(30,34)(32,38)(36,39)(37,40);(1,3)(2,7)(4,8)(5,6)(9,12,14,10)(11,15,13,16)(17,20)(18,22)(19,23)(21,24)(25,28)(26,32)(27,35)(29,36)(30,37)(31,38)(33,39)(34,40);(1,2)(3,7)(4,6)(5,8)(9,13,14,11)(10,15,12,16)(25,29,30,26)(27,31,34,33)(28,36,37,32)(35,38,40,39)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4981000);