#5014830

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

1420 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)(11,12)(13,15)(14,16)(17,18,22,20)(19,23,21,24)(25,26,30,31)(27,28,32,36)(29,33,37,38)(34,35,39,40);(9,11,14,13)(10,12,16,15)(17,19,22,21)(18,23,20,24)(26,31)(27,32)(29,37)(35,40);(9,11,14,13)(10,15,16,12)(17,20,22,18)(19,23,21,24)(25,27)(26,28)(29,34)(30,32)(31,36)(33,35)(37,39)(38,40);(1,2)(3,6)(4,5)(7,8)(9,12,14,15)(10,11,16,13)(25,28)(26,32)(27,31)(29,35)(30,36)(33,39)(34,38)(37,40);(1,3,4,7)(2,6,5,8)(9,13,14,11)(10,15,16,12)(17,21,22,19)(18,23,20,24)(25,29)(26,33)(27,34)(28,35)(30,37)(31,38)(32,39)(36,40)

Polycyclic presentation:

2^9|a^2=h,b^2=fgh,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, 4922026);