#109062

order	512
nilpotency class	3
isoclinism	isoclinic class of #8794
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#382
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₄⋊#95
prime radical	C₂⁴	#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

547 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)(6,11)(7,13)(10,15)(14,16)(17,18)(19,27,30,23)(20,31)(21,35)(22,25)(24,33)(26,36)(28,42,32,46)(29,38,44,41)(34,45)(37,43,39,47)(40,48);(1,3)(2,6,8,14)(4,11)(5,10)(7,9,12,15)(13,16)(17,19)(18,23)(20,32)(21,29)(22,30)(24,39)(25,27)(26,38)(28,34)(31,42)(33,47)(35,41)(36,44)(37,40)(43,48)(45,46);(3,10)(6,14)(9,15)(11,16)(17,20,21,33,22,34,36,48)(18,24,26,31,25,40,35,45)(19,28,29,43,30,32,44,47)(23,37,38,46,27,39,41,42)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 16258);