#3336343

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

1868 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)(5,6,8,10)(7,9,11,12)(13,14,17,20)(15,21,22,18)(16,24,25,19)(23,26,28,27)(29,30,34,32)(31,35,37,38)(33,36,42,40)(39,43,41,44);(5,7,8,11)(6,9,10,12)(15,22)(18,21)(23,28)(26,27)(30,32)(31,37)(36,40)(39,41);(5,7)(6,9)(8,11)(10,12)(13,14)(15,18)(16,19)(17,20)(21,22)(23,26)(24,25)(27,28)(29,31)(30,35)(32,38)(33,41)(34,37)(36,44)(39,42)(40,43);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,15)(14,18)(16,23)(17,22)(19,26)(20,21)(24,27)(25,28)(29,32,34,30)(31,38,37,35)(33,36,42,40)(39,43,41,44);(13,16,17,25)(14,19,20,24)(15,23,22,28)(18,26,21,27)(29,33)(30,36)(31,39)(32,40)(34,42)(35,43)(37,41)(38,44)

Polycyclic presentation:

2^9|a^2=gh,b^2=fh,d^2=f,e^2=g,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, 3243539);