#4311630

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

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fh,c^2=g,d^2=h,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, 4218826);