#315313

order	512
nilpotency class	3
isoclinism	isoclinic class of #288787
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

520 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ef,c^2=hi,d^2=e,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=di,f^c=fi,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, 222509);