#108688

order	512
nilpotency class	3
isoclinism	isoclinic class of #108682
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1434
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

430 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,d^2=i,e^2=hi,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gi,f^b=fhi,e^c=ei,f^c=fh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15884);