#404879

order	512
nilpotency class	3
isoclinism	isoclinic class of #395174
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

1117 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)(12,20)(13,24)(18,33)(19,27)(21,28)(23,31)(25,30)(26,35)(29,36)(32,34);(5,7,9,20,11,19,10,21)(6,12,14,17,15,28,16,27)(8,18,25,36,24,34,23,35)(13,29,30,33,22,26,31,32);(5,8,9,25,11,24,10,23)(6,13,14,30,15,22,16,31)(7,18,20,36,19,34,21,35)(12,26,17,32,28,29,27,33);(2,4)(6,14,15,16)(7,19)(12,27,28,17)(13,30,22,31)(18,34)(20,21)(26,33,29,32)(35,36)

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,d^b=di,e^c=ei,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, 312075);