#141771

order	512
nilpotency class	4
isoclinism	isoclinic class of #141751
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1957
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#493
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

334 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 114 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)(3,9)(4,6)(5,13)(7,11)(8,12)(10,15)(14,16)(17,18)(19,27,30,40)(20,32)(21,24)(22,36)(23,29,37,33)(25,28)(26,35)(31,46,45,41)(34,47)(38,42,48,44)(39,43);(2,6)(3,10)(8,13)(11,16)(17,19,28,42,35,45,20,33,21,30,43,44,22,31,34,29)(18,23,32,46,36,48,25,40,24,37,47,41,26,38,39,27);(1,3,5,11,4,10,12,16)(2,7,8,9,6,14,13,15)(18,24)(23,37)(25,39)(26,36)(27,40)(32,47)(38,48)(41,46)

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,c^2=ei,d^2=g,e^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,c^b=ch,d^b=di,f^b=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48967);