#417783

order	512
nilpotency class	3
isoclinism	isoclinic class of #417778
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷×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

2716 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)(9,10,16,13)(11,21,25,33)(12,26,29,34)(14,20,15,19)(17,24,30,23)(18,31,27,28)(22,36,38,32)(35,39,40,37);(1,2,3,4)(5,6)(7,8)(9,11,14,23,16,25,15,24)(10,17,19,21,13,30,20,33)(12,22,28,37,29,38,31,39)(18,32,26,35,27,36,34,40);(6,8)(9,12)(10,18)(11,22)(13,27)(14,28)(15,31)(16,29)(17,32)(19,26)(20,34)(21,35)(23,37)(24,39)(25,38)(30,36)(33,40);(2,4)(9,13,16,10)(11,21,25,33)(12,27,29,18)(14,19,15,20)(17,24,30,23)(22,35,38,40)(26,31,34,28)(32,39,36,37)

Polycyclic presentation:

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

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, 324979);