#427304

order	512
nilpotency class	3
isoclinism	isoclinic class of #417544
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
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

1394 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)(6,11)(9,21)(12,25)(13,28)(14,22)(16,23)(18,29)(19,34)(24,27)(26,33)(30,36)(31,35);(5,6)(7,12)(8,18)(9,22)(10,14)(11,21)(13,19)(15,30)(16,24)(17,27)(20,28)(23,25)(26,31)(29,34)(32,33)(35,36);(5,7,10,17)(6,12,14,27)(8,15,20,32)(9,23,21,16)(11,24,22,25)(13,26,29,36)(18,33,28,30)(19,35,34,31);(2,4)(5,8)(6,13,14,29)(7,15)(9,19)(10,20)(11,18,22,28)(12,26,27,36)(16,31)(17,32)(21,34)(23,35)(24,33,25,30)

Polycyclic presentation:

2^9|c^2=i,d^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,e^c=ei,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, 334500);