#112190

order	512
nilpotency class	3
isoclinism	isoclinic class of #112177
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	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

729 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)(3,13,15,25,17,8,16,24)(4,18,21,9,22,28,20,27)(14,26,30,23,31,32,19,29)(33,34)(35,45)(36,50,54,59)(37,43)(38,42)(39,44)(40,47)(41,52,60,53)(46,55,62,64)(48,57)(49,56)(51,63,61,58);(1,3)(2,8)(4,19)(5,15)(6,16)(7,17)(9,26)(10,13)(11,24)(12,25)(14,20)(18,23)(21,31)(22,30)(27,32)(28,29)(33,35,39,49)(34,40,44,57)(36,51,54,61)(37,48,38,47)(41,58,60,63)(42,56,43,45)(46,52,62,53)(50,55,59,64);(1,4,7,22)(2,9,10,27)(3,14,17,31)(5,21,6,20)(8,23,13,29)(11,28,12,18)(15,19,16,30)(24,26,25,32)(33,36,39,54)(34,41,44,60)(35,46,49,62)(37,53,38,52)(40,55,57,64)(42,50,43,59)(45,58,56,63)(47,51,48,61)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=i,e^2=h,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,d^b=di,f^b=fh,g^b=ghi,d^c=di,e^c=eh,f^c=fh,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, 19386);