#120911

order	512
nilpotency class	3
isoclinism	isoclinic class of #120908
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#382
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1442
prime radical	C₂⁴	#125
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

366 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)(3,13,15,25,16,8,20,23)(4,17,18,24,19,26,21,9)(14,28,30,27,31,32,29,22)(33,34)(35,44)(36,48)(37,53)(38,42)(39,43)(40,46)(41,51)(45,58)(47,56)(49,63)(50,59)(52,62)(54,57)(55,64)(60,61);(1,3,5,15,7,16,6,20)(2,8,11,23,10,13,12,25)(4,14,18,29,19,31,21,30)(9,22,17,32,24,27,26,28)(33,35)(34,40)(36,49,51,64)(37,54)(38,46)(39,47)(41,58,48,60)(42,56)(43,57)(44,53)(45,52,61,50)(55,59,63,62);(1,4)(2,9,10,24)(3,14)(5,18)(6,21)(7,19)(8,22,13,27)(11,17,12,26)(15,29)(16,31)(20,30)(23,32,25,28)(33,36,37,50,39,51,38,52)(34,41,42,59,43,48,53,62)(35,45,46,49,47,61,54,64)(40,55,44,60,57,63,56,58)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28107);