#110553

order	512
nilpotency class	3
isoclinism	isoclinic class of #110532
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

554 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)(3,13)(4,9,19,25)(5,12)(6,10)(7,11)(8,15)(14,27,29,32)(16,22)(17,23)(18,26,20,24)(21,30,31,28)(33,34,38,42,39,43,37,50)(35,44,46,40,47,55,51,54)(36,41,48,57,49,58,52,56)(45,59,62,63,61,64,60,53);(1,3,6,16,7,17,5,15)(2,8,12,23,11,22,10,13)(4,14,20,30,19,29,18,28)(9,21,26,32,25,31,24,27)(33,35)(34,40)(36,45)(37,51)(38,46)(39,47)(41,53)(42,55)(43,54)(44,50)(48,62)(49,61)(52,60)(56,64)(57,59)(58,63);(1,4,7,19)(2,9)(3,14,17,29)(5,18,6,20)(8,21)(10,24)(11,25)(12,26)(13,27)(15,28,16,30)(22,31)(23,32)(33,36,37,48,39,49,38,52)(34,41,42,56,43,58,50,57)(35,45,51,62,47,61,46,60)(40,53,55,64,54,63,44,59)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17749);