#508529

order	512
nilpotency class	4
isoclinism	isoclinic class of #508008
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 537 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) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415725);