#416202

order	512
nilpotency class	3
isoclinism	isoclinic class of #395174
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2535
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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=eh,d^2=h,f^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,d^c=di,f^c=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, 323398);