#3334932

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2360 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)(9,10,14,11)(12,17,19,15)(13,16,22,18)(20,24,21,23)(25,26)(27,33)(28,34)(29,37)(30,31)(32,40)(35,36)(38,39);(5,6,7,8)(10,11)(12,19)(16,18)(20,21)(26,31)(27,34)(32,37)(35,38);(5,6)(7,8)(9,11,14,10)(12,17,19,15)(13,16,22,18)(20,23,21,24)(25,27)(26,28)(29,38)(30,34)(31,33)(32,39)(35,40)(36,37);(1,2)(3,4)(9,12)(10,15)(11,17)(13,21)(14,19)(16,24)(18,23)(20,22)(25,28)(26,27)(29,39)(30,33)(31,34)(32,38)(35,37)(36,40);(9,13)(10,16)(11,18)(12,20)(14,22)(15,23)(17,24)(19,21)(25,29)(26,32)(27,35)(28,36)(30,40)(31,37)(33,39)(34,38)

Polycyclic presentation:

2^9|a^2=fg,b^2=fh,c^2=fg,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, 3242128);