#418815

order	512
nilpotency class	3
isoclinism	isoclinic class of #417692
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3013	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1942 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)(9,10)(11,20,24,31)(12,17)(13,18)(14,19)(15,22,16,23)(21,33,40,38)(25,32,39,30)(26,34)(27,35)(28,36)(29,37);(5,6)(7,8)(9,11)(10,15)(12,25)(13,23)(14,24)(16,19)(17,33)(18,31)(20,37)(21,26)(22,29)(27,39)(28,40)(30,34)(32,36)(35,38);(2,4)(6,8)(9,11)(10,16)(12,21)(13,22)(14,24)(15,19)(17,32)(18,31)(20,37)(23,29)(25,26)(27,40)(28,39)(30,35)(33,36)(34,38);(9,12,14,27)(10,17,19,35)(11,21,24,40)(13,28,29,26)(15,30,16,32)(18,36,37,34)(20,38,31,33)(22,39,23,25)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 326011);