#149014

order	512
nilpotency class	4
isoclinism	isoclinic class of #135678
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1835
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11268
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

431 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)(3,16,21,44)(4,22)(5,12)(6,31)(7,13)(8,15)(9,18,37,28)(10,25)(11,29)(14,33)(17,49,52,34)(19,38,55,36)(20,35,54,48)(23,57,59,39)(24,42)(26,56)(27,46)(30,45)(32,43)(40,60)(41,58)(47,53,64,51)(50,63,61,62);(1,3,5,28,7,20,29,55,8,21,30,18,33,54,6,19)(2,9,31,16,14,38,45,48,15,37,11,44,13,36,12,35)(4,23,58,53,26,52,32,61,27,59,24,51,60,17,25,50)(10,39,40,62,42,34,46,64,43,57,56,63,41,49,22,47);(1,4,5,24,7,26,29,25,8,27,30,58,33,60,6,32)(2,10,11,40,14,42,12,46,15,43,31,56,13,41,45,22)(3,17,18,50,20,23,19,53,21,52,28,61,54,59,55,51)(9,34,16,47,38,57,48,62,37,49,44,64,36,39,35,63)

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=d,d^2=h,e^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,c^b=cf,d^b=di,g^b=gi,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56210);