#511523

order	512
nilpotency class	4
isoclinism	isoclinic class of #509065
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	C₂²×D₁₆	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

516 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 138 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=efi,d^2=efi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gh,h^b=hi,g^c=gh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 418719);