#317205

order	512
nilpotency class	3
isoclinism	isoclinic class of #317195
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

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

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,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, 224401);