#3362260

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

1628 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)(15,21)(16,22)(17,25)(18,19)(20,28)(23,24)(26,27)(29,30,34,37)(31,38,32,35)(33,36,42,41)(39,44,40,43);(6,10)(11,12)(14,19)(15,22)(20,25)(23,26)(29,31,34,32)(30,35,37,38)(33,39,42,40)(36,43,41,44);(5,6,9,10)(7,11,8,12)(13,15)(14,16)(17,26)(18,22)(19,21)(20,27)(23,28)(24,25)(29,32,34,31)(30,35,37,38)(33,39,42,40)(36,44,41,43);(1,2)(3,4)(5,7,9,8)(6,11,10,12)(13,16)(14,15)(17,27)(18,21)(19,22)(20,26)(23,25)(24,28)(29,30)(31,38)(32,35)(33,41)(34,37)(36,42)(39,43)(40,44);(5,8,9,7)(6,12,10,11)(13,17)(14,20)(15,23)(16,24)(18,28)(19,25)(21,27)(22,26)(29,33)(30,36)(31,39)(32,40)(34,42)(35,43)(37,41)(38,44)

Polycyclic presentation:

2^9|a^2=fg,b^2=fg,c^2=fh,d^2=gh,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, 3269456);