#434473

order	512
nilpotency class	3
isoclinism	isoclinic class of #420064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2420
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

960 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)(7,9)(8,14,25,32)(10,17)(11,18)(12,16)(13,20)(15,26)(19,34,35,23)(21,27)(22,36,30,29)(24,33,28,31);(5,7)(6,13)(8,22)(9,17)(10,15)(11,21)(12,20)(14,23)(16,27)(18,26)(19,24)(25,30)(28,35)(29,31)(32,34)(33,36);(5,8,12,28,11,25,10,24)(6,14,16,31,18,32,17,33)(7,19,20,22,21,35,15,30)(9,23,13,29,27,34,26,36);(2,4)(5,9,10,26,11,27,12,13)(6,15,16,7,18,20,17,21)(8,23,24,36,25,34,28,29)(14,30,31,19,32,22,33,35)

Polycyclic presentation:

2^9|a^2=eg,c^2=ef,d^2=e,e^2=i,g^2=i,h^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,e^b=ei,f^b=fi,g^b=gi,h^b=hi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 341669);