#456503

order	512
nilpotency class	3
isoclinism	isoclinic class of #456501
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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

1174 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 363699);