#488630

order	512
nilpotency class	4
isoclinism	isoclinic class of #488549
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
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2922 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

752 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 130 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)(19,30)(20,24)(21,29)(22,28)(23,26)(25,33)(27,34)(31,45)(32,44)(35,42)(36,41)(37,40)(38,43)(39,48)(46,47);(1,2)(3,6)(4,7)(5,8)(9,13)(10,12)(11,14)(15,16)(18,24)(20,34)(21,22)(25,39)(26,40)(27,29)(28,41)(30,44)(32,47)(33,38)(35,48)(36,37)(42,45)(43,46);(1,3)(2,6)(4,9)(5,10)(7,13)(8,12)(11,15)(14,16)(17,19)(18,25)(20,35)(21,38)(22,33)(23,31)(24,39)(26,43)(27,44)(28,45)(29,30)(32,37)(34,48)(36,47)(40,46)(41,42);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,13,12,16)(18,26)(19,31)(24,40)(27,41)(28,29)(32,35)(33,38)(47,48)

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 395826);