#441435

order	512
nilpotency class	3
isoclinism	isoclinic class of #428059
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
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

789 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)(5,6,11,15,10,16,9,12)(7,13,18,26,19,28,22,24)(8,14,23,25,21,29,20,30)(17,31,32,36,33,27,34,35);(6,12,16,15)(8,20,21,23)(13,26,28,24)(14,29)(17,32,33,34)(25,30)(27,31)(35,36);(5,7,11,18,10,19,9,22)(6,13,12,24,16,28,15,26)(8,17,20,34,21,33,23,32)(14,27,25,35,29,31,30,36);(2,4)(5,8)(6,14)(7,17)(9,23)(10,21)(11,20)(12,25)(13,27)(15,30)(16,29)(18,34)(19,33)(22,32)(24,35)(26,36)(28,31)

Polycyclic presentation:

2^9|a^2=efg,b^2=g,c^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,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, 348631);