#107401

order	512
nilpotency class	3
isoclinism	isoclinic class of #107386
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#3444	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

690 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)(3,12,15,8)(4,17)(5,20)(6,11)(7,10)(9,18)(13,28,30,32)(14,23,16,27)(19,25)(21,24)(22,29,26,31)(33,34,38,43,39,42,37,51)(35,40,47,54,45,55,46,58)(36,48,50,41,49,56,52,57)(44,59,60,53,61,63,62,64);(1,3,6,16,7,15,5,14)(2,8,11,23,10,12,20,27)(4,13,18,29,19,30,21,31)(9,22,25,28,24,26,17,32)(33,35)(34,40,42,55)(36,44)(37,47)(38,46)(39,45)(41,53,57,64)(43,58,51,54)(48,63,56,59)(49,61)(50,62)(52,60);(1,4)(2,9)(3,13)(5,21)(6,18)(7,19)(8,22)(10,24)(11,25)(12,26)(14,31)(15,30)(16,29)(17,20)(23,28)(27,32)(33,36)(34,41)(35,44)(37,52)(38,50)(39,49)(40,53)(42,57)(43,56)(45,61)(46,62)(47,60)(48,51)(54,63)(55,64)(58,59)

Polycyclic presentation:

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

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, 14597);