#3466412

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

2192 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=fgh,d^2=fgi,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, 3373608);