#108919

order	512
nilpotency class	3
isoclinism	isoclinic class of #108898
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₄⋊#95
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#125
LCS	G ▷ C₄² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

463 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ei,d^2=hi,e^2=h,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,d^b=dh,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, 16115);