#508555

order	512
nilpotency class	4
isoclinism	isoclinic class of #508008
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

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

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

normal subgroups only

This group has 442 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)(3,5)(4,6)(7,8)(10,11)(14,15)(17,18,20,23)(19,24,28,35)(21,25,30,36)(22,26,31,37)(27,39,46,38)(29,40,34,44)(32,43,45,41)(33,42,47,48);(1,3,4,7)(2,5,6,8)(9,10,13,11)(12,14,16,15)(18,23)(19,27)(21,32)(22,31)(24,38)(25,41)(28,46)(29,47)(30,45)(33,34)(35,39)(36,43)(40,42)(44,48);(2,6)(5,8)(9,11,13,10)(12,14,16,15)(17,19,20,28)(18,24,23,35)(21,33,30,47)(22,34,31,29)(25,42,36,48)(26,44,37,40)(27,45,46,32)(38,43,39,41);(9,12,13,16)(10,14,11,15)(18,23)(19,28)(25,36)(26,37)(27,46)(29,34)(33,47)(41,43)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=e,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, 415751);