#107876

order	512
nilpotency class	3
isoclinism	isoclinic class of #107858
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	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

709 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:

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

Polycyclic presentation:

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

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, 15072);