#455221

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1300 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)(9,10,16,20)(11,15,19,14)(12,21,27,23)(13,22,31,35)(17,26,33,25)(18,29,36,30)(24,32,40,37)(28,38,39,34);(9,11,14,10,16,19,15,20)(12,23,25,17,27,21,26,33)(13,22,29,18,31,35,30,36)(24,38,28,32,40,34,39,37);(2,4)(9,12,16,27)(10,17,20,33)(11,21,19,23)(13,28,31,39)(14,26,15,25)(18,34,36,38)(22,37,35,32)(24,30,40,29);(6,8)(9,13,15,30,16,31,14,29)(10,18,11,22,20,36,19,35)(12,24,26,39,27,40,25,28)(17,32,23,38,33,37,21,34)

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=i,d^2=ei,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,g^b=gi,d^c=de,e^c=ei,f^c=fi,h^c=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, 362417);