#484961

order	512
nilpotency class	4
isoclinism	isoclinic class of #484199
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

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

normal subgroups only

This group has 725 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)(18,23)(20,30)(21,33)(24,39)(25,38)(26,42)(28,44)(29,43)(31,35)(32,47)(34,36)(37,40)(41,48)(45,46);(1,3)(2,6)(4,9)(5,10)(7,12)(8,13)(11,16)(14,15)(17,18,22,25)(19,24,27,40)(20,31,34,42)(21,35,33,26)(23,36,38,30)(28,45,32,48)(29,46,43,41)(37,44,39,47);(1,4,5,11)(2,7,8,14)(3,9,10,16)(6,12,13,15)(17,19,22,27)(18,24,25,40)(20,32,34,28)(21,29,33,43)(23,37,38,39)(26,41,35,46)(30,47,36,44)(31,48,42,45);(19,27)(24,40)(28,32)(29,43)(37,39)(41,46)(44,47)(45,48)

Polycyclic presentation:

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

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, 392157);