#107985

order	512
nilpotency class	3
isoclinism	isoclinic class of #107966
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

687 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15181);