#228523

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³

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	C₂×C₄×D₄
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

552 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)(6,8)(9,10,14,18)(11,15,20,24)(12,17,13,16)(19,22,21,23)(25,26,29,33,30,27,31,35)(28,32,37,39,36,34,38,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,21)(13,19)(14,20)(16,22)(17,23)(18,24)(25,27,29,35,30,26,31,33)(28,34,37,40,36,32,38,39);(1,3,4,7)(2,6,5,8)(10,16)(12,13)(15,22)(17,18)(19,21)(23,24)(25,28)(26,32)(27,34)(29,37)(30,36)(31,38)(33,39)(35,40);(2,5)(6,8)(9,12,14,13)(10,17,18,16)(11,19,20,21)(15,22,24,23)(26,27)(28,36)(33,35)(37,38)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 135719);