#489097

order	512
nilpotency class	4
isoclinism	isoclinic class of #488901
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#532	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

671 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 671 conjugacy classes of subgroups — too many to draw. Only its 148 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:

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

Polycyclic presentation:

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

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, 396293);