#108507

order	512
nilpotency class	3
isoclinism	isoclinic class of #108504
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#376
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

424 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)(3,11,14,15)(4,7)(5,9)(6,10)(8,13,16,12)(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);(2,7,10,9)(8,15,16,11)(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);(1,3,4,12,6,14,5,13)(2,8,9,11,10,16,7,15)(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=hi,e^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,f^b=fi,f^c=fh

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, 15703);