#363126

order	512
nilpotency class	3
isoclinism	isoclinic class of #25140
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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	C₂×#380
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1001 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=efg,c^2=g,d^2=ef,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,e^b=ei,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,h^d=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, 270322);