#110095

order	512
nilpotency class	3
isoclinism	isoclinic class of #110086
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	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁵	C₄²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

571 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,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,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, 17291);