#231377

order	512
nilpotency class	3
isoclinism	isoclinic class of #30396
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, prime radical	C₂³	C₂×C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

496 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)(3,6)(4,5)(7,8)(9,10)(11,18)(12,16)(13,22)(14,17)(15,20)(19,23)(21,24)(25,26,31,35,29,34,30,39)(27,33,28,38,36,32,37,40);(2,5)(3,7)(25,27,31,28,29,36,30,37)(26,32,35,40,34,33,39,38);(9,11,12,19,14,21,13,20)(10,15,22,24,17,23,16,18)(25,28,31,36,29,37,30,27)(26,33,39,40,34,32,35,38);(1,3,4,7)(2,6,5,8)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,18,23,24)(26,34)(27,36)(28,37)(35,39)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=fh,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,c^b=ci,d^b=de,d^c=di

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, 138573);