#426019

order	512
nilpotency class	3
isoclinism	isoclinic class of #417540
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2598
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1012 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 333215);