#484547

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	#60770	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#408 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 815 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)(21,32)(24,36)(25,40)(26,37)(28,43)(29,42)(31,34)(33,35)(38,47)(39,48)(41,46)(44,45);(1,3)(2,6)(4,9)(5,10)(7,12)(8,13)(11,16)(14,15)(17,18,22,26)(19,24,27,38)(20,31,33,40)(21,34,32,25)(23,35,37,30)(28,44,41,48)(29,45,42,39)(36,46,47,43);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,16)(13,15)(17,19)(18,24)(20,28)(21,29)(22,27)(23,36)(25,39)(26,38)(30,43)(31,44)(32,42)(33,41)(34,45)(35,46)(37,47)(40,48);(19,27)(24,38)(28,41)(29,42)(36,47)(39,45)(43,46)(44,48)

Polycyclic presentation:

2^9|a^2=h,b^2=i,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, 391743);