#127664

order	512
nilpotency class	4
isoclinism	isoclinic class of #11757
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#99
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂×#117	C₂³
socle	C₂²	C₂×#342
prime radical	C₂⁵	C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

613 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)(3,10)(4,8)(5,14)(6,9)(7,11)(12,15)(13,16)(17,18,21,27,23,29,22,28)(19,30,32,46,34,40,33,47)(20,26,36,45,37,25,38,44)(24,39,41,35,43,48,42,31);(1,3)(2,7,9,16)(4,13)(5,11)(6,12)(8,10,14,15)(17,19,22,33,23,34,21,32)(18,24,28,42,29,43,27,41)(20,35,36,48,37,31,38,39)(25,30,44,46,26,40,45,47);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,16,10)(17,20)(18,25)(19,31)(21,36)(22,38)(23,37)(24,40)(26,29)(27,44)(28,45)(30,43)(32,39)(33,48)(34,35)(41,47)(42,46)

Polycyclic presentation:

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

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, 34860);