#120989

order	512
nilpotency class	3
isoclinism	isoclinic class of #120987
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1442
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

406 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:

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

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, 28185);