#450371

order	512
nilpotency class	3
isoclinism	isoclinic class of #418769
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

1041 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=ef,f^2=i,g^2=i,h^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=de,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,g^d=gi,h^d=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, 357567);