#4609293

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1706 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)(5,6,8,10)(7,11,12,9)(14,18)(19,24)(20,25)(26,28)(29,30)(31,37)(32,41)(33,42)(34,35)(36,40)(38,43)(39,44);(5,6,8,10)(7,9,12,11)(15,21)(19,24)(22,27)(26,28)(30,35)(32,42)(36,37)(38,44);(5,6)(7,9)(8,10)(11,12)(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28)(29,31,34,40)(30,36,35,37)(32,38,42,44)(33,43,41,39);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,15,17,21)(14,19,18,24)(16,22,23,27)(20,26,25,28)(29,32)(30,33)(31,38)(34,42)(35,41)(36,43)(37,39)(40,44);(5,7,8,12)(6,9,10,11)(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)(29,33)(30,32)(31,39)(34,41)(35,42)(36,44)(37,38)(40,43)

Polycyclic presentation:

2^9|a^2=fi,b^2=h,c^2=f,d^2=fgh,e^2=fg,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, 4516489);