#459710

order	512
nilpotency class	3
isoclinism	isoclinic class of #459487
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×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	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

883 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:

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

Polycyclic presentation:

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

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, 366906);