#4928682

order	512
nilpotency class	2
isoclinism	isoclinic class of #4861280
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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1637 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)(4,8)(5,7)(9,10,13,16)(11,12,15,14)(17,18,22,24)(19,20,23,21)(25,26)(27,33)(28,30)(29,36)(31,34)(32,37)(35,38)(39,40);(1,3,5,8)(2,6,7,4)(9,11,13,15)(10,14,16,12)(17,19)(18,21)(20,24)(22,23)(26,28)(27,34)(32,36)(35,40);(1,3)(2,4)(5,8)(6,7)(9,12)(10,15)(11,16)(13,14)(17,20,22,21)(18,23,24,19)(25,27)(26,31)(28,33)(29,35)(30,34)(32,38)(36,39)(37,40);(1,3)(2,4)(5,8)(6,7)(9,12)(10,11)(13,14)(15,16)(17,19)(18,20)(21,24)(22,23)(25,28,30,26)(27,33,34,31)(29,32,37,36)(35,38,40,39);(1,4)(2,3)(5,6)(7,8)(9,10)(11,14)(12,15)(13,16)(17,21,22,20)(18,23,24,19)(25,29,30,37)(26,32,28,36)(27,35,34,40)(31,38,33,39)

Polycyclic presentation:

2^9|a^2=gh,b^2=fghi,c^2=hi,d^2=fgh,e^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=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, 4835878);