#317073

order	512
nilpotency class	3
isoclinism	isoclinic class of #317063
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 523 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) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 224269);