#122106

order	512
nilpotency class	3
isoclinism	isoclinic class of #122103
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1446
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

546 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)(3,12)(4,16,19,31)(5,20)(6,10)(7,11)(8,14)(9,18,26,17)(13,22,29,25)(15,23)(21,24)(27,28,30,32)(33,34,38,44,39,43,37,42)(35,45,49,55,47,40,48,56)(36,50,52,58,51,59,53,41)(46,60,61,54,62,64,63,57);(1,3)(2,8,11,24)(4,13)(5,14)(6,21)(7,15)(9,25,26,22)(10,12,20,23)(16,30,31,27)(17,28)(18,32)(19,29)(33,35)(34,40,43,45)(36,46)(37,48)(38,49)(39,47)(41,57,58,54)(42,55,44,56)(50,60,59,64)(51,62)(52,61)(53,63);(1,4)(2,9,11,26)(3,13)(5,17)(6,18)(7,19)(8,22,24,25)(10,16,20,31)(12,27,23,30)(14,28)(15,29)(21,32)(33,36,39,51)(34,41)(35,46,47,62)(37,52,38,53)(40,54)(42,50)(43,58)(44,59)(45,57)(48,61,49,63)(55,64)(56,60)

Polycyclic presentation:

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

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, 29302);