#128121

order	512
nilpotency class	4
isoclinism	isoclinic class of #128009
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#342
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂×C₄×Q₈	C₂³
prime radical	C₂⁴	#95
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂×C₄×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

535 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,d^2=i,e^2=g,f^2=i,b^a=bd,c^a=ci,d^a=dg,f^a=fhi,d^b=dh,e^b=egi,h^b=hi,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35317);