#453957

order	512
nilpotency class	3
isoclinism	isoclinic class of #453948
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3010	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2205 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 361153);