#318548

order	512
nilpotency class	3
isoclinism	isoclinic class of #318531
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2537
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³	#547
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

517 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 517 conjugacy classes of subgroups — too many to draw. Only its 142 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=egh,c^2=i,d^2=eh,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,d^c=dh,f^c=fi,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, 225744);