#364087

order	512
nilpotency class	3
isoclinism	isoclinic class of #184831
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator, prime radical	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1242 conjugacy classes of subgroups; only the 266 normal subgroups are shown.

normal subgroups only

This group has 1242 conjugacy classes of subgroups — too many to draw. Only its 266 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)(3,7)(4,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18,21,26,22,25,34,39)(19,23,29,37,28,36,44,47)(20,30,31,40,32,24,33,38)(27,41,43,48,42,35,45,46);(2,5)(6,7)(9,11,12,16)(10,13,14,15)(17,19)(18,23)(20,27)(21,29)(22,28)(24,35)(25,36)(26,37)(30,41)(31,43)(32,42)(33,45)(34,44)(38,46)(39,47)(40,48);(11,16)(13,15)(17,20,22,32)(18,24,25,30)(19,27,28,42)(21,33,34,31)(23,35,36,41)(26,40,39,38)(29,45,44,43)(37,48,47,46);(1,3,4,8)(2,6,5,7)(10,14)(13,15)(19,28)(23,36)(27,42)(29,44)(35,41)(37,47)(43,45)(46,48)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 271283);