#459882

order	512
nilpotency class	3
isoclinism	isoclinic class of #453980
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	#2782
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

914 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:

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

Polycyclic presentation:

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

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, 367078);