#317337

order	512
nilpotency class	3
isoclinism	isoclinic class of #317315
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

456 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 118 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)(3,8)(4,15)(5,10)(6,11)(7,12)(9,13)(14,16)(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);(1,3,5,14,7,4,6,13)(2,8,10,16,12,15,11,9)(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);(1,4,7,3)(2,9,12,16)(5,14,6,13)(8,10,15,11)(20,32)(23,37)(24,40)(27,41)(28,44)(31,46)(33,39)(45,47);(1,5,7,6)(2,10,12,11)(3,13,4,14)(8,9,15,16)(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=ei,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gh,c^b=cg,d^b=di,f^b=fh,d^c=dh,f^c=fi,g^c=gi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 224533);