#112010

order	512
nilpotency class	3
isoclinism	isoclinic class of #112003
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

786 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)(3,13,15,27,16,8,21,23)(4,17,20,9,18,26,19,25)(14,28,30,22,29,32,31,24)(33,34)(35,44)(36,48,50,57)(37,51)(38,42)(39,43)(40,46)(41,49,58,53)(45,60,63,54)(47,55)(52,56)(59,62,64,61);(1,3)(2,8)(4,14)(5,15)(6,21)(7,16)(9,24)(10,13)(11,23)(12,27)(17,28)(18,29)(19,30)(20,31)(22,25)(26,32)(33,35,39,47)(34,40,43,56)(36,45,50,63)(37,52,38,46)(41,54,58,60)(42,55,51,44)(48,59,57,64)(49,62,53,61);(1,4)(2,9)(3,14)(5,20)(6,19)(7,18)(8,22)(10,25)(11,26)(12,17)(13,24)(15,31)(16,29)(21,30)(23,28)(27,32)(33,36)(34,41)(35,45)(37,49)(38,53)(39,50)(40,54)(42,57)(43,58)(44,59)(46,61)(47,63)(48,51)(52,62)(55,64)(56,60)

Polycyclic presentation:

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

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, 19206);