#117666

order	512
nilpotency class	3
isoclinism	isoclinic class of #117661
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	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

529 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)(4,18,22,9)(5,12)(6,10)(7,11)(8,15)(14,29,31,26)(16,24)(17,25)(19,32,30,23)(20,27,21,28)(33,34,38,44,39,43,37,42)(35,40,46,54,47,55,52,59)(36,48,50,41,51,58,49,57)(45,60,61,56,62,64,63,53);(1,3,5,15,7,17,6,16)(2,8,10,13,11,24,12,25)(4,19,20,14,22,30,21,31)(9,26,27,23,18,29,28,32)(33,35)(34,40)(36,45)(37,52)(38,46)(39,47)(41,56)(42,59)(43,55)(44,54)(48,60)(49,63)(50,61)(51,62)(53,57)(58,64);(1,4)(2,9)(3,14)(5,21)(6,20)(7,22)(8,23)(10,28)(11,18)(12,27)(13,26)(15,19)(16,30)(17,31)(24,32)(25,29)(33,36,39,51)(34,41,43,57)(35,45,47,62)(37,50,38,49)(40,53,55,56)(42,58,44,48)(46,63,52,61)(54,64,59,60)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=i,d^2=h,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,c^b=cf,e^b=ei,d^c=dh,e^c=ei,f^c=fh,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, 24862);