#404878

order	512
nilpotency class	3
isoclinism	isoclinic class of #395173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2482
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1139 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)(7,17)(8,22)(9,16)(10,14)(11,15)(12,19)(13,24)(18,33)(20,27)(21,28)(23,31)(25,30)(26,34)(29,36)(32,35);(5,7,9,19,11,21,10,20)(6,12,14,17,15,27,16,28)(8,18,25,36,24,35,23,34)(13,29,30,33,22,26,31,32);(5,8,9,25,11,24,10,23)(6,13,14,30,15,22,16,31)(7,18,19,36,21,35,20,34)(12,26,17,32,27,29,28,33);(2,4)(6,14,15,16)(12,17,27,28)(13,30,22,31)(26,32,29,33)

Polycyclic presentation:

2^9|b^2=ef,c^2=ef,d^2=g,e^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,e^c=ei,f^c=fi

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, 312074);