#416203

order	512
nilpotency class	3
isoclinism	isoclinic class of #396117
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2420
prime radical	C₂×#3067	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1302 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=eh,d^2=h,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^b=gi,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, 323399);