#441643

order	512
nilpotency class	3
isoclinism	isoclinic class of #425091
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

881 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)(5,6,11,16,10,15,25,28)(7,12,17,30,18,29,35,36)(8,19,21,34,22,13,26,32)(9,23,20,31,24,14,27,33);(5,7,11,17,10,18,25,35)(6,12,28,36,15,29,16,30)(8,20,21,24,22,27,26,9)(13,31,34,23,19,33,32,14);(5,8,10,22)(6,13)(7,9,18,24)(11,26,25,21)(12,23)(14,29)(15,19)(16,34)(17,27,35,20)(28,32)(30,33)(31,36);(2,4)(5,9,10,24)(6,14,15,23)(7,8,18,22)(11,27,25,20)(12,13,29,19)(16,31,28,33)(17,26,35,21)(30,34,36,32)

Polycyclic presentation:

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

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, 348839);