#139470

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	C₂×#540	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#540 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 463 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) = 128

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,d^2=ghi,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,g^a=gi,h^a=hi,d^b=dh,e^b=eghi,f^b=fhi,g^b=gi,h^b=hi,d^c=dghi,e^c=ei,f^c=fghi,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, 46666);