#118212

order	512
nilpotency class	3
isoclinism	isoclinic class of #118207
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

511 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:

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

Polycyclic presentation:

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

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, 25408);