#472077

order	512
nilpotency class	3
isoclinism	isoclinic class of #455107
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

978 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 978 conjugacy classes of subgroups — too many to draw. Only its 202 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)(16,22,34,38,33,40,32,39)(28,41,43,36,42,37,31,44);(1,2,3,4)(5,7,8,11)(6,9,10,12)(13,15,18,29,19,30,20,17)(14,21,24,27,26,35,25,23)(16,31,32,42,33,43,34,28)(22,37,38,44,40,41,39,36);(2,4)(13,16,19,33)(14,22,26,40)(15,28,30,42)(17,31,29,43)(18,34,20,32)(21,36,35,44)(23,37,27,41)(24,39,25,38);(6,10)(9,12)(13,17,20,30,19,29,18,15)(14,23,25,35,26,27,24,21)(16,28,34,43,33,42,32,31)(22,36,39,41,40,44,38,37)

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^c=de,e^c=ei,g^c=gi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 379273);