#491251

order	512
nilpotency class	4
isoclinism	isoclinic class of #491245
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#19329	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

444 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 126 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)(3,6)(4,7)(5,9)(8,12)(10,14)(11,13)(15,16)(17,18,23,27)(19,28,32,36)(20,25,33,40)(21,31,37,38)(22,30,24,26)(29,42,43,47)(34,45,46,48)(35,44,39,41);(1,3)(2,6)(4,11)(5,10)(7,13)(8,15)(9,14)(12,16)(17,19,28,27,22,31,21,30,23,32,36,18,24,38,37,26)(20,29,42,40,35,45,34,44,33,43,47,25,39,48,46,41);(2,7)(4,12)(6,13)(8,9)(11,16)(14,15)(17,20)(18,25)(19,29)(21,34)(22,35)(23,33)(24,39)(26,41)(27,40)(28,42)(30,44)(31,45)(32,43)(36,47)(37,46)(38,48);(3,10)(6,14)(11,16)(13,15)(20,33)(25,40)(29,43)(34,46)(35,39)(41,44)(42,47)(45,48)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398447);