#443566

order	512
nilpotency class	3
isoclinism	isoclinic class of #417464
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1620 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1620 conjugacy classes of subgroups — too many to draw. Only its 210 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) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 350762);