#110503

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	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

556 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)(35,39,44,54)(36,46)(37,49)(38,41)(40,48)(42,51)(43,57,59,60)(45,56,50,53)(47,61)(52,63,64,58)(55,62);(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)(34,39,41,54)(36,43)(37,50)(38,44)(40,52,55,64)(42,56,49,53)(45,51)(46,60,61,57)(47,59)(48,63)(58,62);(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)(33,36,37,48,38,47,51,62)(34,40,42,46,41,55,49,61)(35,43,50,63,44,59,45,58)(39,52,56,60,54,64,53,57)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,e^2=h,f^2=hi,g^2=hi,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, 17699);