#420898

order	512
nilpotency class	3
isoclinism	isoclinic class of #417901
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×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×#585	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1525 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:

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

Polycyclic presentation:

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

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, 328094);