#139674

order	512
nilpotency class	4
isoclinism	isoclinic class of #139568
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

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	#28174	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 358 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) = 128

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46870);