#142237

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	#29738	C₂
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#408 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 755 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)(14,20)(16,29)(17,18)(21,35)(22,30)(23,36)(24,37)(25,41)(26,42)(27,34)(31,43)(32,33)(38,39)(40,44);(9,12)(13,14,19,24)(15,25,28,40)(16,30,33,36)(17,23,18,22)(20,32,37,29)(21,34,39,27)(26,35,31,38)(41,43,44,42);(2,4)(5,7)(6,9)(8,11)(10,12)(13,15)(14,21)(16,31)(17,34)(18,27)(19,28)(20,35)(22,40)(23,25)(24,39)(26,33)(29,43)(30,44)(32,42)(36,41)(37,38)

Polycyclic presentation:

2^9|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, 49433);