#110867

order	512
nilpotency class	3
isoclinism	isoclinic class of #110864
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1437
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

549 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efh,e^2=hi,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,g^a=gi,d^b=di,e^b=eh,f^b=fh,d^c=dh

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, 18063);