#4442933

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
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

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fh,c^2=fghi,d^2=gi,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, 4350129);