#216070

order	512
nilpotency class	3
isoclinism	isoclinic class of #24384
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

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	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

852 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=deh,c^b=ce,d^c=di,f^c=fh,g^c=gh

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, 123266);