#441641

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

913 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)(5,6,11,15,10,14,19,26)(7,12,17,28,8,18,21,30)(9,22,24,31,23,13,25,29)(16,32,33,36,20,27,34,35);(5,7,11,17,10,8,19,21)(6,12,26,30,14,18,15,28)(9,16,24,33,23,20,25,34)(13,27,31,36,22,32,29,35);(5,8)(6,12,14,18)(7,10)(9,20)(11,17)(13,27,22,32)(15,30,26,28)(16,23)(19,21)(24,33)(25,34)(29,36,31,35);(2,4)(5,9,10,23)(6,13,14,22)(7,16,8,20)(11,25,19,24)(12,27,18,32)(15,31,26,29)(17,34,21,33)(28,36,30,35)

Polycyclic presentation:

2^9|a^2=g,b^2=efg,c^2=ef,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, 348837);