#470280

order	512
nilpotency class	3
isoclinism	isoclinic class of #459488
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2684
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

1213 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)(7,13)(8,21)(9,12)(10,16)(11,15)(14,23)(17,30)(18,31)(19,29)(20,26)(22,28)(24,27)(25,32)(33,36)(34,35);(6,12,16,15)(7,17)(13,26,30,29)(14,27,32,21)(18,33)(19,20)(22,34)(28,31,35,36);(2,4)(5,7,10,17)(6,13,16,30)(8,22,24,34)(9,20,11,19)(12,26,15,29)(14,31,32,36)(18,25,33,23)(21,28,27,35);(5,8,11,25,10,24,9,23)(6,14,15,21,16,32,12,27)(7,18,20,34,17,33,19,22)(13,28,26,31,30,35,29,36)

Polycyclic presentation:

2^9|b^2=gi,c^2=i,d^2=eg,e^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,d^c=de,e^c=ei,h^c=hi,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, 377476);