#411620

order	512
nilpotency class	3
isoclinism	isoclinic class of #396117
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2537
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1806 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318816);