#244582

order	512
nilpotency class	3
isoclinism	isoclinic class of #244547
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

956 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 956 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,d^2=h,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,g^a=ghi,c^b=cg,e^b=eh,f^b=fh,g^b=gi,e^c=ei,f^c=fh,g^c=gi,e^d=eh,f^d=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151778);