#108684

order	512
nilpotency class	3
isoclinism	isoclinic class of #108682
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1434
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#126
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

396 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)(3,13,16,8)(4,18,20,26)(5,11)(6,12)(7,10)(9,19,25,21)(14,28)(15,24,17,23)(22,29)(27,30)(31,32)(33,34)(35,44,48,59)(36,49)(37,42)(38,52)(39,43)(40,46,55,47)(41,50)(45,54,61,64)(51,56)(53,57)(58,62,63,60);(1,3,6,17,7,16,5,15)(2,8,12,24,10,13,11,23)(4,14,21,30,20,31,19,29)(9,22,26,28,25,27,18,32)(33,35)(34,40,43,55)(36,45)(37,46)(38,47)(39,48)(41,54,57,64)(42,44,52,59)(49,63,56,58)(50,60)(51,61)(53,62);(1,4)(2,9,10,25)(3,14)(5,19)(6,21)(7,20)(8,22,13,27)(11,18,12,26)(15,29)(16,31)(17,30)(23,32,24,28)(33,36,37,50,39,51,38,53)(34,41,42,56,43,57,52,49)(35,45,46,60,48,61,47,62)(40,54,44,58,55,64,59,63)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15880);