#4418793

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=fh,d^2=g,e^2=f,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, 4325989);