#120876

order	512
nilpotency class	3
isoclinism	isoclinic class of #120874
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	C₄⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

384 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,d^2=h,e^2=h,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,c^b=cf,d^b=di,f^b=fi,g^c=gi

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, 28072);