#4973723

order	512
nilpotency class	2
isoclinism	isoclinic class of #4932388
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

1448 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fh,c^2=fi,d^2=gh,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg

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, 4880919);