#430461

order	512
nilpotency class	3
isoclinism	isoclinic class of #418484
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
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

1214 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)(5,6,10,16,12,15,11,7)(8,19,23,28,21,17,22,13)(9,18,26,14,24,29,25,30)(20,33,34,31,35,27,36,32);(5,7,10,15,12,16,11,6)(8,19,22,28,21,17,23,13)(9,18,25,14,24,29,26,30)(20,31,34,33,35,32,36,27);(5,8)(6,13)(7,17)(9,20)(10,23)(11,22)(12,21)(14,27)(15,28)(16,19)(18,31)(24,35)(25,36)(26,34)(29,32)(30,33);(2,4)(5,9,11,25,12,24,10,26)(6,14,7,18,15,30,16,29)(8,20,22,36,21,35,23,34)(13,27,17,31,28,33,19,32)

Polycyclic presentation:

2^9|a^2=ef,b^2=f,d^2=efi,f^2=i,h^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,g^b=gi,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, 337657);