#378114

order	512
nilpotency class	3
isoclinism	isoclinic class of #23886
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1441 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:

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

Polycyclic presentation:

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

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, 285310);