#476653

order	512
nilpotency class	3
isoclinism	isoclinic class of #476095
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

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	#23399	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1036 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 383849);