#5014701

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

1434 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=gh,c^2=fgh,d^2=g,e^2=fghi,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, 4921897);