#317229

order	512
nilpotency class	3
isoclinism	isoclinic class of #317219
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

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

normal subgroups only

This group has 497 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,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=fi,g^a=gh,c^b=cg,d^b=di,f^b=fh,d^c=di,f^c=fi,g^c=gh

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, 224425);