#107569

order	512
nilpotency class	3
isoclinism	isoclinic class of #107562
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	C₂⁴×C₄	C₂³
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

897 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2)(3,9)(4,12,13,7)(5,8)(6,11)(10,14,16,15)(17,18,22,28,23,26,21,27)(19,29,31,42,32,24,37,39)(20,33,35,25,36,41,34,40)(30,44,45,38,46,48,47,43);(1,3)(2,6,8,9)(4,10)(5,11)(7,14,12,15)(13,16)(17,19)(18,24)(20,30)(21,31)(22,37)(23,32)(25,38)(26,29)(27,39)(28,42)(33,48)(34,45)(35,47)(36,46)(40,43)(41,44);(1,4)(2,7)(3,10)(5,13)(6,14)(8,12)(9,15)(11,16)(17,20,23,36)(18,25,26,40)(19,30,32,46)(21,35,22,34)(24,38,29,43)(27,41,28,33)(31,47,37,45)(39,44,42,48)

Polycyclic presentation:

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

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, 14765);