#4386255

order	512
nilpotency class	2
isoclinism	isoclinic class of #4379698
automorphism	Aut(G) order 2097152
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

1758 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)(7,11)(9,12)(14,18)(16,23)(19,24)(21,27)(29,30,34,32)(31,37,38,35)(33,40,42,36)(39,43,41,44);(5,6,8,10)(7,9,11,12)(14,18)(19,24)(20,25)(26,28)(30,32)(31,38)(36,40)(39,41);(1,2)(3,4)(13,14)(15,19)(16,20)(17,18)(21,26)(22,24)(23,25)(27,28)(29,31,34,38)(30,35,32,37)(33,41,42,39)(36,44,40,43);(5,7)(6,9)(8,11)(10,12)(13,15,17,22)(14,19,18,24)(16,21,23,27)(20,26,25,28)(29,32,34,30)(31,37,38,35)(33,40,42,36)(39,44,41,43);(13,16,17,23)(14,20,18,25)(15,21,22,27)(19,26,24,28)(29,33)(30,36)(31,39)(32,40)(34,42)(35,43)(37,44)(38,41)

Polycyclic presentation:

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

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, 4293451);