#421910

order	512
nilpotency class	3
isoclinism	isoclinic class of #421159
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1146 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 329106);