#4494800

order	512
nilpotency class	2
isoclinism	isoclinic class of #4465744
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1669 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)(3,8)(4,5)(6,7)(9,10)(11,12)(13,14)(15,16)(17,18)(19,22)(20,21)(23,24)(25,26,31,29)(27,32,28,35)(30,33,38,37)(34,39,36,40);(1,3)(2,5)(4,6)(7,8)(10,14)(12,15)(18,20)(19,23)(25,27,31,28)(26,32,29,35)(30,34,38,36)(33,39,37,40);(2,7)(3,4)(9,11,13,16)(10,12,14,15)(17,18,21,20)(19,24,23,22)(25,28,31,27)(26,32,29,35)(30,36,38,34)(33,39,37,40);(1,4,6,3)(2,8,7,5)(9,11)(10,15)(12,14)(13,16)(17,19,21,23)(18,22,20,24)(25,29,31,26)(27,32,28,35)(30,37,38,33)(34,39,36,40);(1,5,6,8)(2,3,7,4)(9,12,13,15)(10,11,14,16)(17,20,21,18)(19,24,23,22)(25,30,31,38)(26,33,29,37)(27,34,28,36)(32,39,35,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gi,d^2=g,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4401996);