#120717

order	512
nilpotency class	3
isoclinism	isoclinic class of #118505
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#388
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1442
prime radical	C₂⁴	#125
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

394 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27913);