#405370

order	512
nilpotency class	3
isoclinism	isoclinic class of #394335
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1292 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=e,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,f^c=fi,g^c=gi

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, 312566);