#317941

order	512
nilpotency class	3
isoclinism	isoclinic class of #317935
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	#537
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

500 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 140 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)(3,13)(4,15)(5,8)(6,12)(7,11)(9,10)(14,16)(17,18,22,26)(19,23,28,38)(20,30,33,42)(21,34,36,25)(24,35,41,32)(27,43,31,48)(29,46,45,39)(37,47,40,44);(2,8,12,11)(3,14,10,15)(4,16)(9,13)(17,19)(18,23)(20,31)(21,29)(22,28)(24,40)(25,39)(26,38)(27,33)(30,48)(32,47)(34,46)(35,44)(36,45)(37,41)(42,43);(1,3)(2,9)(4,8)(5,14)(6,10)(7,15)(11,16)(12,13)(17,20,22,33)(18,24,26,41)(19,27,28,31)(21,35,36,32)(23,37,38,40)(25,42,34,30)(29,47,45,44)(39,48,46,43);(1,4,5,9,6,16,7,13)(2,10,11,15,12,3,8,14)(19,28)(23,38)(27,31)(29,45)(37,40)(39,46)(43,48)(44,47)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 225137);