#120430

order	512
nilpotency class	3
isoclinism	isoclinic class of #120426
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1441
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#125
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

382 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=d,d^2=h,e^2=hi,f^2=hi,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,f^b=fhi,e^c=ei,f^c=fhi,g^c=gh

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, 27626);