#4057708

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

1494 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fgi,c^2=gh,d^2=fgh,e^2=fg,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, 3964904);