#4435191

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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

1802 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)(5,6,8,10)(7,11,12,9)(14,18)(15,21)(16,23)(26,28)(30,34)(31,37)(32,39)(43,44);(14,18)(19,24)(20,25)(26,28)(29,30,33,34)(31,35,37,41)(32,36,39,42)(38,43,40,44);(5,7,8,12)(6,9,10,11)(13,14,17,18)(15,19,21,24)(16,20,23,25)(22,26,27,28)(29,30)(31,35)(32,36)(33,34)(37,41)(38,43)(39,42)(40,44);(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,15)(14,19)(16,22)(17,21)(18,24)(20,26)(23,27)(25,28)(29,31)(30,35)(32,40)(33,37)(34,41)(36,44)(38,39)(42,43);(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)(29,32)(30,36)(31,38)(33,39)(34,42)(35,43)(37,40)(41,44)

Polycyclic presentation:

2^9|a^2=gh,b^2=f,c^2=fg,d^2=gi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=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, 4342387);