#128182

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×#117	C₂³
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

544 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)(3,11,13,7)(4,8)(5,10)(6,9)(12,16,14,15)(17,18,21,26,22,24,34,39)(19,27,29,43,30,42,33,47)(20,25,31,40,32,38,46,48)(23,35,37,28,36,44,41,45);(1,3,5,14,6,13,4,12)(2,7,10,16,9,11,8,15)(17,19)(18,23)(20,28,32,45)(21,33)(22,30)(24,36)(25,27,38,42)(26,41)(29,34)(31,35,46,44)(37,39)(40,47,48,43);(1,4,6,5)(2,8,9,10)(3,12,13,14)(7,15,11,16)(18,24)(23,36)(25,38)(26,39)(27,42)(37,41)(40,48)(43,47)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,d^2=i,e^2=i,f^2=g,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, 35378);