#107957

order	512
nilpotency class	3
isoclinism	isoclinic class of #107946
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	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

694 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,e^2=i,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,d^b=di,e^b=eh,f^b=fh,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, 15153);