#141549

order	512
nilpotency class	4
isoclinism	isoclinic class of #141503
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1972
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₈	C₂×C₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1451 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

284 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 76 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=e,d^2=ghi,e^2=gh,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,g^a=gi,c^b=cg,d^b=dh,e^b=ei,f^b=fh,f^c=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48745);