#4862350

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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1828 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fgh,c^2=f,d^2=f,e^2=fg,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, 4769546);