#112454

order	512
nilpotency class	3
isoclinism	isoclinic class of #112446
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

736 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)(3,13,17,23,15,8,16,24)(4,9,20,26,19,27,21,28)(14,22,18,25,30,29,31,32)(33,34)(35,44)(36,49,53,41)(37,54)(38,42)(39,43)(40,46)(45,61,63,57)(47,59)(48,56)(50,55,62,64)(51,58,52,60);(1,3)(2,8)(4,18)(5,17)(6,16)(7,15)(9,25)(10,13)(11,24)(12,23)(14,20)(19,31)(21,30)(22,26)(27,32)(28,29)(33,35,39,47)(34,40,43,56)(36,50,53,62)(37,48,38,46)(41,57,49,61)(42,59,54,44)(45,51,63,52)(55,60,64,58);(1,4)(2,9)(3,14)(5,21)(6,20)(7,19)(8,22)(10,27)(11,28)(12,26)(13,29)(15,30)(16,31)(17,18)(23,32)(24,25)(33,36)(34,41)(35,45)(37,52)(38,51)(39,53)(40,55)(42,60)(43,49)(44,61)(46,50)(47,63)(48,62)(54,58)(56,64)(57,59)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 19650);