#142239

order	512
nilpotency class	4
isoclinism	isoclinic class of #142236
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#447
commutator, Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#408 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

726 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 78 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 — 3 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=gi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,f^a=fh,g^a=gi,c^b=cf,d^b=dg,f^b=fi,g^b=gi,d^c=dh,f^c=fi

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, 49435);