#5014711

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

1564 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)(11,15)(12,13)(14,16)(17,18,23,24)(19,22,21,20)(25,26,30,28)(27,33,34,31)(29,36,37,32)(35,38,40,39);(10,12)(14,15)(17,19,23,21)(18,20,24,22)(27,34)(31,33)(35,40)(38,39);(9,11)(10,14)(12,15)(13,16)(17,20,23,22)(18,19,24,21)(25,27)(26,31)(28,33)(29,35)(30,34)(32,38)(36,39)(37,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,15)(13,16)(17,21,23,19)(18,20,24,22)(25,28,30,26)(27,31,34,33)(29,36,37,32)(35,38,40,39);(1,3,4,7)(2,6,5,8)(9,12,13,10)(11,15,16,14)(17,22,23,20)(18,19,24,21)(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=fgh,c^2=fgh,d^2=fg,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, 4921907);