#112412

order	512
nilpotency class	3
isoclinism	isoclinic class of #112405
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

753 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)(3,11,14,7)(4,15,18,8)(5,10)(6,9)(12,22)(13,27,19,23)(16,24)(17,28,20,25)(21,26)(29,30)(31,32)(33,34,38,44,39,43,37,42)(35,45,48,40,49,56,47,55)(36,50,51,58,52,41,53,57)(46,60,61,54,62,59,63,64);(1,3)(2,7)(4,16)(5,19)(6,14)(8,24)(9,11)(10,27)(12,17)(13,21)(15,32)(18,31)(20,30)(22,25)(23,26)(28,29)(33,35,39,49)(34,40,43,55)(36,46,52,62)(37,48,38,47)(41,54,50,64)(42,56,44,45)(51,63,53,61)(57,60,58,59);(1,4)(2,8)(3,12)(5,20)(6,18)(7,22)(9,15)(10,28)(11,29)(13,31)(14,30)(16,19)(17,21)(23,32)(24,27)(25,26)(33,36)(34,41)(35,46)(37,51)(38,53)(39,52)(40,54)(42,57)(43,50)(44,58)(45,59)(47,61)(48,63)(49,62)(55,64)(56,60)

Polycyclic presentation:

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

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, 19608);