#442333

order	512
nilpotency class	3
isoclinism	isoclinic class of #421903
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
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

869 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:

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

Polycyclic presentation:

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

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, 349529);