#425860

order	512
nilpotency class	3
isoclinism	isoclinic class of #417784
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1064 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:

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

Polycyclic presentation:

2^9|b^2=f,c^2=ef,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=dei,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=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, 333056);