#110499

order	512
nilpotency class	3
isoclinism	isoclinic class of #110476
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	#117
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

538 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,e^2=h,f^2=hi,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,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, 17695);