#147306

order	512
nilpotency class	4
isoclinism	isoclinic class of #12952
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1864
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10908
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

509 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2)(3,14,17,45)(4,20,23,35)(5,26)(6,11)(7,13)(8,16,33,53)(9,22,36,21)(10,29)(12,31)(15,47)(18,46,19,38)(24,39,25,59)(27,42,54,34)(28,37,57,43)(30,40)(32,51)(41,60)(44,48)(49,56)(50,64)(52,61)(55,63)(58,62);(1,3,6,18,7,17,31,19)(2,8,11,34,13,33,12,42)(4,15,24,50,23,51,25,55)(5,27,29,53,30,54,60,16)(9,32,37,63,36,47,43,64)(10,38,40,14,41,46,26,45)(20,56,39,61,35,44,59,62)(21,52,28,48,22,58,57,49);(1,4,6,24,7,23,31,25)(2,9,12,43,13,36,11,37)(3,15,18,50,17,51,19,55)(5,28,29,22,30,57,60,21)(8,32,42,64,33,47,34,63)(10,39,26,20,41,59,40,35)(14,44,38,61,45,56,46,62)(16,52,27,48,53,58,54,49)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fh,d^2=h,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,d^b=dhi,e^b=ehi,f^b=fi,h^b=hi,d^c=dhi,e^c=eh,f^c=fi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314268, #72314278
maximal normal subgroup of	#72314268, #72314278

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54502);