#434082

order	512
nilpotency class	3
isoclinism	isoclinic class of #419792
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
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

1152 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=e,c^2=efi,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^c=gi,h^c=hi,e^d=ei,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, 341278);