#108476

order	512
nilpotency class	3
isoclinism	isoclinic class of #106004
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	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

452 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:

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

Polycyclic presentation:

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

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, 15672);