#141796

order	512
nilpotency class	4
isoclinism	isoclinic class of #12525
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#493
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1970
prime radical	C₂³	#466
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

300 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 90 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfi,c^2=ehi,d^2=g,e^2=hi,f^2=g,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=cgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72210421
maximal normal subgroup of	#72210421
maximal quotient of	#72210358

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48992);