#108105

order	512
nilpotency class	3
isoclinism	isoclinic class of #108094
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	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

665 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)(35,43)(36,47)(37,50)(38,42)(39,45)(40,48)(41,52)(44,59)(46,54)(49,56)(51,55)(53,63)(57,61)(58,62)(60,64);(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)(34,39,42,43)(36,44)(37,46)(38,45)(40,53,55,57)(41,54,50,58)(47,64,56,59)(48,61)(49,60)(51,63)(52,62);(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)(33,36,38,49)(34,40,42,55)(35,44,45,60)(37,51,52,48)(39,53,43,57)(41,56,50,47)(46,63,62,61)(54,59,58,64)

Polycyclic presentation:

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

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, 15301);