#326494

order	512
nilpotency class	3
isoclinism	isoclinic class of #326487
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2563
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

437 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 437 conjugacy classes of subgroups — too many to draw. Only its 138 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=eghi,d^2=eh,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,f^c=fi,g^c=gh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 233690);