#1539633

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
automorphism	?
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

2341 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)(9,10,14,15)(11,16,17,12)(13,18,21,22)(19,23,24,20)(25,26)(27,31)(28,34)(29,37)(30,33)(32,38)(35,40)(36,39);(6,8)(10,15)(11,17)(13,21)(20,23)(25,27,30,28)(26,31,33,34)(29,35,38,36)(32,39,37,40);(5,6)(7,8)(9,11)(10,16)(12,15)(13,19)(14,17)(18,23)(20,22)(21,24)(25,28,30,27)(26,31,33,34)(29,36,38,35)(32,39,37,40);(1,2)(3,4)(9,12,14,16)(10,17,15,11)(13,20,21,23)(18,24,22,19)(25,26)(27,34)(28,31)(29,32)(30,33)(35,40)(36,39)(37,38);(9,13)(10,18)(11,19)(12,20)(14,21)(15,22)(16,23)(17,24)(25,29)(26,32)(27,35)(28,36)(30,38)(31,39)(33,37)(34,40)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=g,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df

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, 1446829);