#472080

order	512
nilpotency class	3
isoclinism	isoclinic class of #455110
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

964 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 964 conjugacy classes of subgroups — too many to draw. Only its 198 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) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=egh,c^2=i,d^2=ei,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=di,d^c=de,e^c=ei,g^c=gi,h^c=hi

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, 379276);