#119848

order	512
nilpotency class	3
isoclinism	isoclinic class of #119843
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
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

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27044);