#424743

order	512
nilpotency class	3
isoclinism	isoclinic class of #419800
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1130 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)(7,17,21,25)(8,14)(9,12,10,15)(13,18,27,20)(19,31)(22,30)(23,26)(24,29)(28,34)(32,36)(33,35);(1,2)(3,4)(6,12)(7,18)(8,22)(9,10)(13,27)(14,29)(15,16)(19,33)(20,21)(23,24)(28,31)(35,36);(2,4)(5,7,9,20,11,21,10,18)(6,13,15,17,16,27,12,25)(8,19,23,34,24,33,22,32)(14,28,30,31,29,36,26,35);(5,8)(6,14)(7,19)(9,23)(10,22)(11,24)(12,26)(13,28)(15,30)(16,29)(17,31)(18,32)(20,34)(21,33)(25,35)(27,36)

Polycyclic presentation:

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

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, 331939);