#139668

order	512
nilpotency class	4
isoclinism	isoclinic class of #139558
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1893
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#28143	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

399 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 399 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,d^2=gi,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,c^b=ci,e^b=eh,d^c=dh,f^c=fh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265732, #72308622
maximal normal subgroup of	#72265732, #72308622

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46864);