#118546

order	512
nilpotency class	3
isoclinism	isoclinic class of #118543
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	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

530 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)(33,34,38,44,39,43,37,42)(35,40,48,56,47,57,46,55)(36,49,53,41,51,59,52,58)(45,60,62,54,50,63,61,64);(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)(36,50,51,45)(37,46)(38,48)(39,47)(41,54,58,64)(42,55)(43,57)(44,56)(49,60,59,63)(52,62,53,61);(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)(34,41)(35,45)(37,53)(38,52)(39,51)(40,54)(42,59)(43,58)(44,49)(46,62)(47,50)(48,61)(55,63)(56,60)(57,64)

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,e^2=hi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,c^b=cf,e^b=eh,f^b=fhi,d^c=di,e^c=ehi,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, 25742);