#116690

order	512
nilpotency class	3
isoclinism	isoclinic class of #116687
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

519 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)(4,18,22,28)(5,11)(6,10)(7,12)(8,15)(9,21,26,20)(14,23,31,32)(16,24)(17,27)(19,29,30,25)(33,34,38,44,39,42,37,43)(35,40,46,56,47,55,53,60)(36,48,51,58,52,41,50,59)(45,61,49,64,62,54,63,57);(1,3,5,15,7,17,6,16)(2,8,10,13,12,24,11,27)(4,19,20,14,22,30,21,31)(9,25,18,23,26,29,28,32)(33,35)(34,40)(36,49)(37,53)(38,46)(39,47)(41,57)(42,55)(43,60)(44,56)(45,50)(48,64)(51,62)(52,63)(54,58)(59,61);(1,4)(2,9)(3,14)(5,20)(6,21)(7,22)(8,23)(10,18)(11,28)(12,26)(13,29)(15,30)(16,19)(17,31)(24,32)(25,27)(33,36,39,52)(34,41,42,48)(35,45,47,62)(37,51,38,50)(40,54,55,61)(43,59,44,58)(46,63,53,49)(56,64,60,57)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 23886);