#108453

order	512
nilpotency class	3
isoclinism	isoclinic class of #105688
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₄×C₂²⋊C₄
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1434
prime radical	C₂⁴	#126
LCS	G ▷ C₄² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

435 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)(3,11)(4,12)(5,8)(6,10)(7,13)(9,15)(14,16)(17,18,22,27,23,28,21,26)(19,29,31,40,32,39,37,24)(20,33,35,25,36,41,34,42)(30,44,45,48,47,38,46,43);(1,3,6,7,5,12,15,16)(2,4,10,14,8,11,9,13)(17,19)(18,24,28,40)(20,30)(21,31)(22,37)(23,32)(25,38,42,44)(26,29,27,39)(33,43,41,48)(34,45)(35,46)(36,47);(1,4,5,11)(2,7,8,16)(3,10,12,9)(6,14,15,13)(17,20,21,34,23,36,22,35)(18,25,26,33,28,42,27,41)(19,30,31,45,32,47,37,46)(24,38,29,43,40,44,39,48)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,d^2=i,e^2=h,f^2=i,g^2=hi,b^a=bd,c^a=ce,e^a=eh,g^a=gi,f^b=fi,e^c=eh

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15649);