#107915

order	512
nilpotency class	3
isoclinism	isoclinic class of #107898
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

682 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)(3,13,15,8,17,24,16,23)(4,18,20,26,21,9,19,25)(14,28,30,22,31,27,29,32)(33,34,38,43,39,44,37,42)(35,45,49,55,47,40,48,56)(36,50,52,41,53,57,51,58)(46,60,63,59,61,64,62,54);(1,3,6,16,7,17,5,15)(2,8,12,24,11,23,10,13)(4,14,20,30,21,31,19,29)(9,22,26,28,18,32,25,27)(33,35)(34,40,44,45)(36,46)(37,48)(38,49)(39,47)(41,54,58,59)(42,55,43,56)(50,64,57,60)(51,62)(52,63)(53,61);(1,4)(2,9)(3,14)(5,19)(6,20)(7,21)(8,22)(10,25)(11,18)(12,26)(13,27)(15,29)(16,30)(17,31)(23,32)(24,28)(33,36,39,53)(34,41,44,58)(35,46,47,61)(37,52,38,51)(40,54,45,59)(42,57,43,50)(48,63,49,62)(55,60,56,64)

Polycyclic presentation:

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

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, 15111);