#420763

order	512
nilpotency class	3
isoclinism	isoclinic class of #417901
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,𝔽₄)
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2684
prime radical	C₂×#3018	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1877 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 327959);