#484975

order	512
nilpotency class	4
isoclinism	isoclinic class of #484217
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3050	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#424 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

708 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 708 conjugacy classes of subgroups — too many to draw. Only its 140 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,e^2=gi,f^2=i,g^2=i,b^a=be,c^a=cf,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

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 392171);