#321988

order	512
nilpotency class	3
isoclinism	isoclinic class of #321983
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	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

649 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 649 conjugacy classes of subgroups — too many to draw. Only its 150 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=eghi,d^2=e,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,d^b=dh,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, 229184);