#127849

order	512
nilpotency class	4
isoclinism	isoclinic class of #11799
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×C₄×Q₈	C₂³
socle	C₂²	C₂×#342
prime radical	C₂⁴	#95
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂×C₄×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

523 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 523 conjugacy classes of subgroups — too many to draw. Only its 128 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=g,e^2=gi,f^2=g,b^a=bd,c^a=ci,d^a=dg,f^a=fgh,c^b=cg,d^b=dh,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, 35045);