#110495

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	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

550 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)(3,11,14,7)(4,15)(5,18)(6,9)(8,17)(10,20)(12,26,27,32)(13,22,19,24)(16,29)(21,28,25,31)(23,30)(33,34,37,42,39,43,38,52)(35,44,47,58,46,40,48,54)(36,41,49,55,51,56,50,62)(45,57,60,64,59,53,61,63);(1,3)(2,7)(4,12)(5,19)(6,14)(8,21)(9,11)(10,24)(13,20)(15,26)(16,27)(17,31)(18,22)(23,25)(28,30)(29,32)(33,35,38,48,39,46,37,47)(34,40,42,54,43,44,52,58)(36,45,50,61,51,59,49,60)(41,53,55,63,56,57,62,64);(1,4,5,17,6,16,20,30)(2,8,10,15,9,23,18,29)(3,12,19,31,14,27,13,28)(7,21,24,26,11,25,22,32)(33,36,37,49,39,51,38,50)(34,41,42,55,43,56,52,62)(35,45,47,60,46,59,48,61)(40,53,54,63,44,57,58,64)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,e^2=h,f^2=i,g^2=i,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, 17691);