#436182

order	512
nilpotency class	3
isoclinism	isoclinic class of #428059
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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

937 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)(7,16,20,30)(8,21,25,12)(9,13)(10,14)(11,19,27,18)(15,23)(17,31)(22,35)(24,29,36,28)(26,32)(33,34);(5,7)(6,11)(8,22)(9,18)(10,20)(12,26)(13,30)(14,27)(15,16)(17,25)(19,23)(21,34)(24,33)(28,35)(29,31)(32,36);(5,8,23,36,10,25,9,24)(6,12,15,29,14,21,13,28)(7,17,18,32,20,22,19,33)(11,26,30,35,27,34,16,31);(2,4)(6,13,14,15)(7,18,20,19)(11,27)(12,28,21,29)(16,30)(17,32,22,33)(26,34)(31,35)

Polycyclic presentation:

2^9|a^2=fg,c^2=ei,d^2=f,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^c=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, 343378);