#3468234

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1785 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fg,d^2=fhi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3375430);