#505571

order	512
nilpotency class	4
isoclinism	isoclinic class of #499897
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

705 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 705 conjugacy classes of subgroups — too many to draw. Only its 144 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)(3,7)(4,6)(5,8)(9,10,12,14)(11,13,15,16)(17,18)(19,23)(20,24)(21,34)(22,27)(25,31)(26,32)(28,36)(29,35)(30,44)(33,40)(37,43)(38,42)(39,45)(41,46)(47,48);(1,3,4,8)(2,5,6,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,23)(20,29)(21,30)(22,28)(24,35)(25,37)(26,38)(27,36)(31,43)(32,42)(33,41)(34,44)(39,47)(40,46)(45,48);(11,15)(13,16)(18,24)(20,31)(21,32)(23,35)(25,26)(27,40)(29,43)(30,42)(33,45)(34,39)(36,46)(37,38)(41,48)(44,47);(2,6)(5,7)(19,28)(23,36)(29,41)(30,42)(35,46)(37,47)(38,44)(43,48)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 412767);