#377547

order	512
nilpotency class	3
isoclinism	isoclinic class of #24398
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#121
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

946 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=fh,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,f^c=fi,g^c=gi,h^c=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 284743);