#321916

order	512
nilpotency class	3
isoclinism	isoclinic class of #321901
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2541
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³	#535
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

634 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 634 conjugacy classes of subgroups — too many to draw. Only its 186 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=eg,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,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, 229112);