#108679

order	512
nilpotency class	3
isoclinism	isoclinic class of #108672
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	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

395 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)(17,18,21,27,23,26,22,28)(19,29,31,41,32,24,35,38)(20,25,34,40,33,39,36,42)(30,43,45,48,44,37,46,47);(2,7,10,9)(8,15,16,11)(17,19,22,31,23,32,21,35)(18,24,28,38,26,29,27,41)(20,30,34,46,33,44,36,45)(25,37,40,48,39,43,42,47);(1,3,4,12,6,14,5,13)(2,8,9,11,10,16,7,15)(17,20)(18,25,26,39)(19,30)(21,36)(22,34)(23,33)(24,37,29,43)(27,42,28,40)(31,46)(32,44)(35,45)(38,48,41,47)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15875);