#417779

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	C₂²×#454	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2708 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)(10,15)(11,19)(13,27)(17,30)(18,31)(21,37)(22,23)(24,34)(25,28)(29,33)(32,36)(35,39);(1,2,3,4)(5,6)(7,8)(9,10,13,15,14,18,27,31)(11,20,22,33,23,16,19,29)(12,24,25,34,26,30,28,17)(21,38,35,32,39,40,37,36);(6,8)(9,11)(10,16)(12,21)(13,22)(14,23)(15,29)(17,32)(18,20)(19,27)(24,40)(25,35)(26,39)(28,37)(30,38)(31,33)(34,36);(2,4)(9,12,14,26)(10,17,18,34)(11,21,23,39)(13,28,27,25)(15,30,31,24)(16,32,20,36)(19,35,22,37)(29,38,33,40)

Polycyclic presentation:

2^9|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, 324975);