#4867107

order	512
nilpotency class	2
isoclinism	isoclinic class of #4861280
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

1912 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(1,2,3,4)(7,11)(10,12)(13,14,17,18)(15,21,22,19)(16,24,26,20)(23,27,25,28)(29,30)(31,36)(32,41)(33,34)(35,38)(37,44)(39,43)(40,42);(6,9)(10,12)(14,18)(15,22)(20,24)(23,25)(29,31,35,32)(30,36,38,41)(33,42,44,39)(34,40,37,43);(5,6)(7,10)(8,9)(11,12)(13,14)(15,19)(16,20)(17,18)(21,22)(23,27)(24,26)(25,28)(29,32,35,31)(30,36,38,41)(33,42,44,39)(34,43,37,40);(1,2)(3,4)(13,15,17,22)(14,19,18,21)(16,25,26,23)(20,28,24,27)(29,33)(30,37)(31,39)(32,42)(34,38)(35,44)(36,40)(41,43);(5,7)(6,10)(8,11)(9,12)(13,16,17,26)(14,20,18,24)(15,23,22,25)(19,27,21,28)(29,34,35,37)(30,33,38,44)(31,40,32,43)(36,42,41,39)

Polycyclic presentation:

2^9|a^2=fgi,b^2=fg,c^2=fg,d^2=f,e^2=g,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, 4774303);