#369175

order	512
nilpotency class	3
isoclinism	isoclinic class of #25440
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1828 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:

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

Polycyclic presentation:

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

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, 276371);