#120732

order	512
nilpotency class	3
isoclinism	isoclinic class of #120728
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

412 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,c^b=cf,f^b=fi,g^b=gh,e^c=eh,f^c=fhi

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, 27928);