#459709

order	512
nilpotency class	3
isoclinism	isoclinic class of #459486
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

893 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)(8,13,21,26,22,31,36,30)(17,28,20,34,33,32,35,29);(1,2)(3,4)(6,11,15,14)(12,27,16,25)(13,26,31,30)(28,34,32,29);(2,4)(5,7)(6,12)(8,20)(9,23)(10,19)(11,25)(13,29)(14,27)(15,16)(17,21)(18,24)(22,35)(26,32)(28,30)(31,34)(33,36);(5,8,9,21,10,22,24,36)(6,13,11,26,15,31,14,30)(7,17,18,20,19,33,23,35)(12,28,27,34,16,32,25,29)

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^c=de,e^c=ei,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 366905);