#107813

order	512
nilpotency class	3
isoclinism	isoclinic class of #107810
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	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

810 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 15009);