#427890

order	512
nilpotency class	3
isoclinism	isoclinic class of #417682
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
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

1239 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:

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

Polycyclic presentation:

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

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, 335086);