#3475029

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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1598 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)(13,14)(15,20)(16,25)(17,21)(18,19)(22,23)(24,27)(26,28)(29,30,34,38)(31,36,32,35)(33,41,42,37)(39,43,40,44);(6,9)(10,12)(14,19)(16,21)(20,27)(22,28)(29,31,34,32)(30,35,38,36)(33,39,42,40)(37,43,41,44);(5,6,8,9)(7,10,11,12)(13,15)(14,20)(16,22)(17,26)(18,24)(19,27)(21,28)(23,25)(29,30)(31,35)(32,36)(33,41)(34,38)(37,42)(39,44)(40,43);(1,2,3,4)(13,16)(14,17)(15,22)(18,21)(19,25)(20,26)(23,27)(24,28)(29,32,34,31)(30,36,38,35)(33,39,42,40)(37,43,41,44);(5,7,8,11)(6,10,9,12)(13,17)(14,21)(15,23)(16,19)(18,25)(20,22)(24,26)(27,28)(29,33,34,42)(30,37,38,41)(31,39,32,40)(35,43,36,44)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=fgi,e^2=gh,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, 3382225);