#406468

order	512
nilpotency class	3
isoclinism	isoclinic class of #395242
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1322 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 313664);