#4311626

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

1754 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=f,d^2=fh,e^2=i,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, 4218822);