#465674

order	512
nilpotency class	3
isoclinism	isoclinic class of #456510
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

1105 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)(5,6,10,13)(7,17,20,26)(8,22,24,30)(9,16,11,15)(12,21,28,18)(14,23,29,25)(19,27,34,36)(31,32,35,33);(1,2,4,3)(5,7,11,21,10,20,9,18)(6,12,16,26,13,28,15,17)(8,19,23,33,24,34,25,32)(14,27,30,35,29,36,22,31);(1,2)(3,4)(6,13)(7,18)(8,23)(9,11)(12,26)(14,22)(17,28)(20,21)(24,25)(27,36)(29,30)(32,33);(1,3,4,2)(5,8,10,24)(6,14,13,29)(7,19,20,34)(9,25,11,23)(12,27,28,36)(15,22,16,30)(17,31,26,35)(18,32,21,33)

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,h^b=hi,d^c=de,e^c=ei,g^c=gi,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, 372870);