#377180

order	512
nilpotency class	3
isoclinism	isoclinic class of #24412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#120
prime radical	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1245 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)(9,10,15,20,16,21,14,35)(11,17,24,38,25,28,36,37)(12,26,31,39,30,23,29,18)(13,19,33,27,34,40,32,22);(1,3,5,4)(2,6,8,7)(9,11)(10,17)(12,27)(13,23)(14,36)(15,24)(16,25)(18,33)(19,29)(20,38)(21,28)(22,30)(26,34)(31,40)(32,39)(35,37);(1,3,5,4)(2,7,8,6)(9,12,14,29,16,30,15,31)(10,18,20,26,21,39,35,23)(11,22,24,19,25,27,36,40)(13,28,32,38,34,17,33,37);(1,4,5,3)(2,7,8,6)(9,13,15,33,16,34,14,32)(10,19,20,27,21,40,35,22)(11,23,24,18,25,26,36,39)(12,28,31,37,30,17,29,38)

Polycyclic presentation:

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

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, 284376);