#491660

order	512
nilpotency class	4
isoclinism	isoclinic class of #491653
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#493
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#3147	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

520 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 150 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=fh,d^2=i,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,h^a=hi,c^b=ci,d^c=dg

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, 398856);