#317204

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

Polycyclic presentation:

2^9|a^2=i,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, 224400);