#3402686

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

1704 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=hi,d^2=g,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, 3309882);