#139464

order	512
nilpotency class	4
isoclinism	isoclinic class of #139440
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1898
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	#25614	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#542 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

472 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 66 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)(3,9)(4,12,14,7)(5,11)(6,8)(10,13,16,15)(17,18,22,27)(19,28,31,44)(20,32)(21,24,36,26)(23,33)(25,35)(29,39)(30,42,46,43)(34,41)(37,47)(38,45)(40,48);(1,3,9,2,6,11,5,8)(4,10,13,7,14,16,15,12)(17,19,20,33)(18,23,32,44)(21,30,34,47)(22,31,35,29)(24,37,41,43)(25,28,27,39)(26,38,40,42)(36,46,48,45);(1,4)(2,7)(3,10)(5,15)(6,14)(8,12)(9,13)(11,16)(18,24)(21,36)(23,37)(25,40)(26,27)(28,42)(30,46)(32,41)(34,48)(38,39)(43,44)(45,47)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,d^2=ghi,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eghi,g^a=gi,h^a=hi,d^b=dh,e^b=egh,f^b=fhi,g^b=gi,h^b=hi,d^c=dgh,e^c=ei,f^c=fgh,f^d=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46660);