#4198619

order	512
nilpotency class	2
isoclinism	isoclinic class of #4131229
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, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1588 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fghi,c^2=fgh,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4105815);