#4310194

order	512
nilpotency class	2
isoclinism	isoclinic class of #4308740
automorphism	Aut(G) order 4194304
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1794 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=f,d^2=f,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^c=df

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, 4217390);