#478427

order	512
nilpotency class	3
isoclinism	isoclinic class of #476609
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 385623);