#488822

order	512
nilpotency class	4
isoclinism	isoclinic class of #488719
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33779	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

694 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 694 conjugacy classes of subgroups — too many to draw. Only its 132 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 — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|c^2=hi,d^2=gh,e^2=gi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dgh,e^a=eg,f^a=fi,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,d^c=di,e^c=eh,f^c=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396018);