#141825

order	512
nilpotency class	4
isoclinism	isoclinic class of #141704
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1961
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#19667	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

307 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 307 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 256

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=egi,d^2=gi,e^2=hi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,c^b=cg,e^b=ei,f^c=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, 49021);