#421909

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	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1216 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=fi,c^2=e,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, 329105);