#107573

order	512
nilpotency class	3
isoclinism	isoclinic class of #107562
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	C₂×#121	C₂×C₄
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:

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

Polycyclic presentation:

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

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, 14769);