#4967467

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, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1758 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)(4,6)(5,7)(9,10)(11,15)(12,13)(14,16)(19,23)(21,24)(25,26,30,28)(27,31,33,34)(29,36,37,32)(35,40,39,38);(1,3,5,4)(2,6,7,8)(10,13)(11,16)(27,33)(31,34)(35,39)(38,40);(3,4)(6,8)(9,11)(10,14)(12,16)(13,15)(17,18,20,22)(19,21,23,24)(25,27,30,33)(26,31,28,34)(29,35,37,39)(32,38,36,40);(1,4,5,3)(2,6,7,8)(17,19)(18,21)(20,23)(22,24)(25,28,30,26)(27,34,33,31)(29,36,37,32)(35,40,39,38);(1,2)(3,8)(4,6)(5,7)(9,10)(11,14)(12,13)(15,16)(25,29)(26,32)(27,35)(28,36)(30,37)(31,38)(33,39)(34,40)

Polycyclic presentation:

2^9|a^2=fh,b^2=fg,c^2=fhi,d^2=gh,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, 4874663);