#110234

order	512
nilpotency class	3
isoclinism	isoclinic class of #9443
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1437
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

560 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)(3,12)(4,16,18,25)(5,19)(6,10)(7,11)(8,14)(9,17,26,21)(13,28,31,22)(15,23)(20,24)(27,30,32,29)(33,34)(35,39,44,51)(36,40,46,53)(37,48)(38,41)(42,49)(43,55)(45,59,58,52)(47,61,60,54)(50,57)(56,62)(63,64);(1,3)(2,8)(4,13)(5,20)(6,14)(7,15)(9,22)(10,23)(11,24)(12,19)(16,27)(17,30)(18,31)(21,29)(25,32)(26,28)(33,35,37,45,38,44,49,58)(34,39,42,52,41,51,48,59)(36,43,47,56,46,57,60,64)(40,50,54,62,53,55,61,63);(1,4,6,21,7,18,5,17)(2,9,10,25,11,26,19,16)(3,13,14,29,15,31,20,30)(8,22,23,32,24,28,12,27)(33,36)(34,40,41,53)(35,43)(37,47)(38,46)(39,50,51,55)(42,54,48,61)(44,57)(45,56)(49,60)(52,62,59,63)(58,64)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=hi,e^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gi,d^b=dh,e^b=ei,f^b=fi,d^c=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, 17430);