#4757448

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
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, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1623 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=fh,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef

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, 4664644);