#224123

order	512
nilpotency class	3
isoclinism	isoclinic class of #24269
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle, prime radical	C₂³	C₂²×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

744 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=h,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,c^b=chi,d^b=dei,f^c=fh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 131319);