#317027

order	512
nilpotency class	3
isoclinism	isoclinic class of #317015
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

618 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 618 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 64

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

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

Polycyclic presentation:

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

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, 224223);