#420899

order	512
nilpotency class	3
isoclinism	isoclinic class of #417902
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

1515 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)(7,9)(8,21,24,33)(10,15)(11,16)(12,17)(13,25,32,29)(14,26)(18,34,35,30)(19,27)(20,28)(22,36,31,23);(1,2)(3,4)(5,7,10,14,12,20,11,19)(6,9,15,26,17,28,16,27)(8,22,25,35,24,31,29,18)(13,30,21,23,32,34,33,36);(2,4)(5,8)(6,13)(7,18)(9,23)(10,29)(11,25)(12,24)(14,31)(15,33)(16,21)(17,32)(19,22)(20,35)(26,30)(27,34)(28,36);(5,9)(6,14)(7,16)(8,23)(10,26)(11,27)(12,28)(13,31)(15,20)(17,19)(18,21)(22,32)(24,36)(25,34)(29,30)(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=dei,e^c=ei,f^c=fi,g^c=gi,g^d=gi,h^d=hi

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, 328095);